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In this video, we will see how to apply the second derivative test to classify a critical point as either a local minimum, a local maximum, or a point of inflection.
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We should already be familiar with the definition of critical points as points of a function where the slope of the tangent to the curve is equal to zero or is undefined and how to find the critical points for function using differentiation.
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You may also be familiar with the first derivative test for classifying critical points, which is also called determining their nature.
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Critical points can be either local minima, local maxima, or points of inflection.
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And they are classified according to the shape of the curve at that point.
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This is determined by the behavior of the slope of the curve around the point.
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Recall that the first derivative of a function π prime of π₯ or dπ¦ by dπ₯, if weβre using Leibnizβs notation, tells us the slope of a curve.
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Thatβs the rate of change of the curve itself.
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And at critical points, the first derivative is equal to zero.
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Therefore, the second derivative of a function, which is the derivative of the first derivative, tells us the slope of the slope.
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Or more usefully, it tells us about the rate of change of the slope of a curve.
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Letβs consider how the slope of a curve changes around a critical point, starting with a local minima.
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By drawing in tangents to the curve either side of the critical point, we see that the slope of the curve and therefore the first derivative of the function is negative to the left of our critical point and is positive to the right of the critical point.
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The slope and therefore the value of π prime of π₯ changes from negative to zero to positive.
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And therefore, the value of the slope is increasing.
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Recall that if a function is increasing, it has a positive derivative.
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So this tells us that as the slope is increasing, the derivative of the slope is positive.
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The derivative of the slope is the second derivative of the original function.
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So we can conclude that at a local minimum, the second derivative of the function will be positive: π double prime of π₯ is greater than zero.
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We can apply the same reasoning to a local maximum.
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This time, the slope π prime of π₯ changes from positive to zero to negative and therefore the value of π prime of π₯ is decreasing.
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If a function is decreasing, then its derivative is negative.
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So we can conclude that at a local maximum, the derivative of π prime of π₯, thatβs π double prime of π₯, the second derivative of the original function, will be negative.
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Unfortunately, the second derivative test is not particularly useful for identifying points of inflection.
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At a point of inflection, the slope either changes from positive to zero to positive or from negative to zero to negative.
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And so, the sign of π prime of π₯ is the same either side of a point of inflection.
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Consequently, we canβt use results about increasing or decreasing functions to use the second derivative test to classify a point of inflection.
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In fact, it turns out that at points of inflection, the second derivative of a function is equal to zero.
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But this can also be true at some local minima or local maxima.
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So it isnβt enough in order to conclude that at a critical point must be a point of inflection.
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For example, consider the function π of π₯ equals π₯ to the power of four.
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We know from its graph that it has a local minimum point at the origin.
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If we find the first derivative π prime of π₯, this is equal to four π₯ cubed.
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And setting this equal to zero, we confirm that there is indeed a critical point when π₯ is equal to zero.
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The second derivative of the function π double prime of π₯ is 12π₯ squared.
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And if we were to substitute π₯ equals zero into the second derivative, we would get zero.
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But as weβve seen, this critical point is a local minimum, not a point of inflection.
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What this tells us is that if the second derivative at a critical point is equal to zero, we must instead use the first derivative test to determine the nature of the critical point because it could be a point of inflection, but it may also be a local minimum or a local maximum.
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Letβs now consider some examples.
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Determine the local maximum and minimum values of the function π¦ equals negative three π₯ squared minus six π₯ minus four.
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First, we recall that at critical points, the first derivative of the function β in this case dπ¦ by dπ₯ β is equal to zero.
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So our first step is going to be to find the first derivative of this function.
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By applying the power rule of differentiation, we find that dπ¦ by dπ₯ is equal to negative six π₯ minus six.
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We then set our expression for dπ¦ by dπ₯ equal to zero and solve for π₯, giving π₯ is equal to negative one.
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Our function, therefore, has one critical point, which occurs when π₯ is equal to negative one.
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Next, we need to evaluate the function at the critical point, which we do by substituting π₯ equals negative one into the equation weβve been given.
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We obtain π¦ equals negative three multiplied by negative one squared minus six multiplied by negative one minus four which simplifies to negative one.
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This tells us then that the only critical point of this function is the point with coordinates negative one, negative one.
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But we need to determine whether this is a local minimum or local maximum, which weβll do by applying the second derivative test.
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To find the second derivative, we need to differentiate our first derivative with respect to π₯.
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So weβre finding the derivative of negative six π₯ minus six with respect to π₯.
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Applying the power rule, we see that this derivative is just equal to negative six.
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Now, this second derivative is actually just a constant because weβve differentiated a quadratic expression twice.
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So we donβt need to substitute the π₯-coordinate at our critical point in in order to evaluate because the second derivative is constant for all values of π₯.
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We note that negative six is less than zero.
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We recall that if the second derivative of a function is negative at the critical point, then the critical point is a local maximum.
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So the point negative one, negative one is indeed a local maximum of this function.
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So we can conclude that this function has no local minimum value but has a local maximum value of negative one.
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Notice that is the value of the function itself that we are giving here, not the π₯-value, although they are both the same in this instance.
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We can also confirm this result using our knowledge of the graphs of quadratic functions.
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As the coefficient of π₯ squared in this curve is negative, the graph of this quadratic will be a negative parabola.
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We know that parabolas have only a single critical point.
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And if the coefficient of π₯ squared is negative, then their critical point will be a local maximum.
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Letβs now consider another example.
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Find the points π₯, π¦ where π¦ equals nine π₯ plus nine over π₯ has a local maximum or a local minimum.
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Local maxima and local minima are examples of critical points.
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And we recall that at the critical points of a function, the first derivative dπ¦ by dπ₯ is equal to zero.
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Before differentiating, we may find it helpful to rewrite the second term in our function as nine π₯ to the negative one.
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We can then use the power rule of differentiation to find the first derivative dπ¦ by dπ₯.
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Remember that when we differentiate, we decrease the power by one.
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So when we decrease that power of negative one, it will become negative two not zero.
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Watch out for that!
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Thatβs a common mistake.
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We can rewrite this derivative as nine minus nine over π₯ squared and weβll then set this derivative equal to zero.
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Weβll now solve the resulting equation in order to find the π₯-values at the critical points.
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We begin by multiplying every term in the equation by π₯ squared.
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We can then divide through by nine to give π₯ squared minus one equals zero.
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Add one to both sides and then finally take the square root, remembering that we have both positive and negative solutions.
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We find that π₯ is equal to positive or negative one.
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So this function has two critical points.
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Next, we need to find the π¦-values at each critical point by evaluating the function itself.
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When π₯ is equal to positive one, π¦ is equal to nine multiplied by one plus nine over one, which is equal to 18, giving a critical point of one, 18.
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When π₯ is equal to negative one, π¦ is equal to negative 18.
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So our second critical point has coordinates negative one, negative 18.
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We now need to determine whether these critical points are local minima or local maxima, which weβll do using the second derivative test.
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Weβll clear some space in order to do this.
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To find the second derivative d two π¦ by dπ₯ squared, we need to differentiate the first derivative, which was nine minus nine π₯ to the power of negative two with respect to π₯.
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Doing so, we obtain negative nine multiplied by negative two π₯ to the power of negative three, which we can write as 18 over π₯ cubed.
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Next, we need to evaluate this second derivative at each of our critical points.
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When π₯ is equal to negative one, the second derivative is 18 over negative one cubed, which is equal to negative 18.
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This is less than zero.
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And we recall that if the second derivative of a function is negative at a critical point, then the critical point is a local maximum.
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Evaluating the second derivative when π₯ is equal to positive one gives 18 over one cubed, which is 18.
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And as this is greater than zero, we conclude that the critical point when π₯ is equal to one is a local minimum.
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So weβve completed the problem.
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We answered that the point one, 18 is a local minimum and the point negative one, negative 18 is a local maximum.
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In our next example, weβll apply our knowledge of the second derivative test for local extrema to a problem involving differentiation of trigonometric functions.
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Find, if any, the local maximum and minimum values of π of π₯ equals 19 sin π₯ plus 15 cos π₯, together with their type.
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We recall first of all that at the critical points of a function, the first derivative π prime of π₯ is equal to zero.
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Weβll also need to recall the cycle that we can use for differentiating sine and cosine.
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π prime of π₯ is, therefore, equal to 19 cos π₯ minus 15 sin π₯ and we set this equal to zero.
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To solve, we can first separate the two terms onto opposite sides of the equation and then divide both sides of the equation by both cos π₯ and 15 to give sin π₯ over cos π₯ is equal to 19 over 15.
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At this point, we recall one of our trigonometric identities: tan π is equal to sin π over cos π.
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So we have tan π₯ equals 19 over 15.
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To solve, we apply the inverse tan function.
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And we must recall at this point that in order to differentiate trigonometric functions, we must be working with the angle measured in radians as the key limits used when we first derive the derivatives from first principles are only true in radians.
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So when we evaluate π₯ on our calculators, we must make sure weβre working in radians.
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We find then that π₯ is equal to 0.9025 radians.
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However, tan π₯ is a periodic function with a period of π.
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So there are other solutions to this equation which will correspond to other critical points of the function π.
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This means that critical points will occur at this π₯-value that weβve just found plus or minus integer multiples of π.
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Adding π to our value of 0.9025 gives 4.0441 radians.
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So this will be the second π₯-value at which a local maximum or minimum occurs.
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Next, we need to evaluate the function π of π₯ at each of the critical points.
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For our first critical point, when π₯ is equal to 0.9025, we get 24.21 to two decimal places.
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And at our second critical point, where π₯ is equal to 4.0441, we get negative 24.21 to two decimal places.
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Now, this makes sense because the sine and cosine functions each have a horizontal line of symmetry on the π₯-axis and therefore so will a sum or difference of the sine and cosine functions, meaning that the absolute value of the local maximum will be the same as the absolute value of the local minimum.
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Now, finally, we need to apply the second derivative test to classify these critical points.
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So letβs make a little bit of room.
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We differentiate π prime of π₯ to give negative 19 sin π₯ minus 15 cos π₯.
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Now we need to evaluate this function at each of our critical points, but thereβs a trick that we can use here.
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As weβve differentiated twice, weβve been halfway around our cycle of differentiation, which means that the second derivative is actually almost identical to the original function.
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Whatβs different is that both terms are negative instead of positive.
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But if we factor this negative one out of our expression, we see that in this instance, π double prime of π₯ is actually equal to negative π of π₯.
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The reason this is useful is because weβve already evaluated π of π₯ at each of our critical points.
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So we can use the values weβve already found in order to determine the second derivative at our critical points.
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At our first critical point, when π₯ is equal to 0.9025, π of π₯ was equal to 24.21.
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So the second derivative π double prime of π₯ will be equal to negative 24.21.
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As this is less than zero, it tells us that this critical point will be a local maximum.
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At our second critical point, the value of π of π₯ was negative 24.21.
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So the value of π double prime of π₯ will be positive 24.21 and as this is greater than nought, our second critical point is a local minimum.
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Now these are local minimum and maximum values.
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But due to the shape of the graph of π of π₯, theyβre also the absolute minimum and maximum values of the function.
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So we can conclude that the local and absolute minimum value of the function is negative 24.21 and the local and absolute maximum value of the function is 24.21.
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Letβs consider our final example.
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Suppose π prime of four equals zero and π double prime of four equals negative four.
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What can you say about π at the point π₯ equals four?
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π has a local minimum at π₯ equals four.
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π has a local maximum at π₯ equals four.
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π has a point of inflection at π₯ equals four.
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It is not possible to state the nature of the turning point of π at π₯ equals four.
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Or π has a vertical tangent at π₯ equals four.
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Letβs take each of the pieces of information weβve been given in turn.
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Firstly, weβre told that π prime of four is equal to zero.
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And if the first derivative of a function is equal to zero at a given point, then the function has a critical point at that point.
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So we know that π has a critical point when π₯ is equal to four.
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Next, weβre told that π double prime of four is equal to negative four.
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So the second derivative of our function π is negative when π₯ is equal to four.
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The second derivative will be negative at a local maximum.
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So we can conclude that π has a local maximum at π₯ equals four.
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Thatβs the second option in the list weβve been given.
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The first, third, and fourth options are therefore false.
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If a point is a local maximum, it canβt also be a local minimum or a point of inflection.
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And it has been possible for us to determine the nature of this turning point.
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Letβs consider the fifth option.
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We know that the first derivative of our function π is zero when π₯ is equal to four, which means that the slope of the curve and the slope of the tangent will be zero.
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Therefore, π will have a horizontal, not a vertical tangent at π₯ equals four.
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So weβve completed the problem.
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π has a local maximum at π₯ equals four.
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Letβs summarize what weβve seen in this video.
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If π is a differentiable function such that the first derivative π prime of π is equal to zero, then π has a critical point at π₯ equals π.
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If the second derivative π double prime of π is positive, then the critical point is a local minimum.
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But if the second derivative π double prime of π is negative, the critical point is a local maximum.
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If the second derivative π double prime of π is equal to zero, then the critical point could be a point of inflection.
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But itβs possible that it could also be a local minimum or a local maximum.
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So in this instance, weβd need to use the first derivative test in order to classify the critical point.