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Find the inflection point of π of π₯ equals π₯ cubed minus 12π₯ minus one.
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First of all, what are inflection points?
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These are points where a curve goes from having an increasing slope, also called concave up, to a decreasing slope, also called concave down, or vice versa.
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So how do we find these points?
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Well, remember that the first derivative of a curve tells us the slope of the curve, and the second derivative tells us if the slope is increasing or decreasing.
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Where the slope is increasing, the first derivative is increasing, and so the second derivative is positive.
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Where the slope is decreasing, the first derivative is decreasing.
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So the second derivative is negative.
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So weβre looking for points where the second derivative goes from being positive to negative or negative to positive.
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In other words, weβre looking for those points where the second derivative is equal to zero.
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So letβs firstly find what the second derivative is.
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Weβll do this by firstly finding the first derivative π prime of π₯.
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And we do this by differentiating π of π₯ with respect to π₯.
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Weβll do this term by term, starting with the first term, that is, π₯ cubed.
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We can do this using the power rule for differentiation, which tells us that the derivative of π₯ to the πth power is ππ₯ to the power of π minus one.
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Essentially, this means we multiply by the power and then subtract one from the power.
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So π₯ cubed differentiates to three π₯ squared.
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So now we differentiate the next term, which is 12π₯.
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So now we differentiate the next term negative 12π₯.
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This just differentiates to negative 12.
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And our last term negative one is just a constant, and constants differentiate to zero.
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So π prime of π₯ is equal to three π₯ squared minus 12.
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To find π double prime of π₯, we differentiate π prime of π₯ with respect to π₯.
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Letβs start with the term three π₯ squared.
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Using the power rule again, we find that three π₯ squared differentiates to six π₯.
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And as negative 12 is just a constant, this differentiates to zero.
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So π double prime of π₯ is six π₯.
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Remember, we said weβre looking for the points where the second derivative is equal to zero.
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So weβre going to set six π₯ equal to zero.
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Solving for π₯, we find that π₯ must be equal to zero.
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So we know we have an inflection point at π₯ equals zero.
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Thatβs the only solution to six π₯ equals zero.
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But letβs find the corresponding π¦-value.
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We can do this by substituting π₯ is equal to zero into our equation for π of π₯.
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This gives us zero cubed minus 12 multiplied by zero minus one, which gives us negative one.
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So we can conclude that the inflection point for this curve is zero, negative one.
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Letβs now draw a very quick sketch of the graph of π₯ cubed minus 12π₯ minus one so we can check our answer.
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Letβs mark on what we found for our point of inflection zero, negative one.
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And in fact, to the left of this point, we can see that the graph is concave down.
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And to the right of this point, we can see that the graph is concave up.
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So zero, negative one is definitely the point of inflection for this curve.