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Determine the intervals over which the function π of π₯ is equal to 11π₯ cubed minus eight π₯ squared is increasing and over which it is decreasing.
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The question gives us a polynomial function π of π₯.
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It wants us to find the intervals over which this function is increasing and the intervals over which this function is decreasing.
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And we recall for a differentiable function π, we say itβs increasing when its derivative is greater than zero.
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Intuitively, this is because its slope will be greater than zero, so its slope is pointing upwards.
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So, itβs getting larger.
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And we also know that for a differentiable function π, we say itβs decreasing when its derivative is less than zero.
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Since our function π of π₯ is a polynomial, itβs differentiable for all real numbers.
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So, we want to find the intervals which contain those values of π₯ such that the derivative of π is greater than zero and the derivative of π is less than zero.
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Letβs start by finding our derivative function π prime of π₯.
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Since weβre differentiating a polynomial, we can do this by using the power rule for differentiation.
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We multiply by the exponent and then reduce the exponent by one.
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Differentiating our polynomial term by term using this rule, we get three times 11 times π₯ to the power of three minus one minus two times eight times π₯ to the power of two minus one, which simplifies to give us 33π₯ squared minus 16π₯.
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So, our derivative function π prime of π₯ is a quadratic.
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And we want to find the values of π₯ where this quadratic is greater than zero and the values of π₯ where this quadratic is less than zero.
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Thereβs a few different methods we could use.
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Weβre going to do this by sketching a graph of π prime of π₯.
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So, we want to sketch a graph of the quadratic π¦ is equal to 33π₯ squared minus 16π₯.
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We see the leading term of our polynomial is 33π₯ squared.
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Since 33 is positive, our sketch should have a similar shape to the parabola π¦ is equal to π₯ squared.
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Next, we can find the values of our π₯-intercepts by factoring this equation.
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We see that both terms share a factor of π₯.
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Taking out the shared factor of π₯, we get π₯ multiplied by 33π₯ minus 16.
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And we can find the π₯-intercepts of our sketch by solving each factor equal to zero.
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This gives us π₯-intercepts of π₯ is equal to zero and π₯ is equal to 16 divided by 33.
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We can now sketch a graph of our derivative function, π¦ is equal to 33π₯ squared minus 16π₯.
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We mark our π₯-intercepts, zero and 16 divided by 33.
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And we know that our sketch of π¦ is equal to π prime of π₯ should have a similar shape to the parabola π¦ is equal to π₯ squared.
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We want to use this sketch to find the values of π₯ where π prime of π₯ is bigger than zero and where π prime of π₯ is less than zero.
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This is the same as finding the values of π₯ where our curve lies above and below the π₯-axis.
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We can see if π₯ is less than zero, our curve lies above the π₯-axis.
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And if π₯ is greater than 16 over 33, our curve also lies above the π₯-axis.
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So, we have when π₯ is less than zero, π prime of π₯ is greater than zero.
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And when π₯ is greater than 16 over 33, π prime of π₯ is also greater than zero.
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Saying π₯ is less than zero is the same as saying that π₯ is in the open interval from negative β to zero.
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And saying π₯ is greater than 16 over 33 is the same as saying π₯ is in the open interval from 16 over 33 to β.
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So, we found the intervals where our function π of π₯ is increasing.
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We can do the same to find the intervals where π of π₯ is decreasing.
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From our sketch, we can see that π prime of π₯ is below the π₯-axis when π₯ is between zero and 16 over 33.
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So, when π₯ is greater than zero and π₯ is less than 16 over 33, we have that π prime of π₯ is less than zero.
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This tells us that our function π of π₯ is decreasing for all values of π₯ on the open interval from zero to 16 over 33.
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Therefore, weβve shown that the function π of π₯ is equal to 11π₯ cubed minus eight π₯ squared is decreasing over the open interval from zero to 16 over 33 and increasing over the open intervals from negative β to zero and from 16 over 33 to β.