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Solve the given problems by finding the appropriate derivatives.Find the ordered pair $(x, y)$ on the graph of $y=x^{3}-3 x^{2}$ for which $f^{\prime \prime}(x)=0 .$ (This is called an inflection point.)

Calculus 1 / AB

Chapter 23

The Derivative

Section 9

Higher Derivatives

Derivatives

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Lectures

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In mathematics, precalculu…

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In mathematics, a function…

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Find the inflection points…

04:41

Find the $x$ -coordinate o…

05:14

Identify the coordinates o…

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Sketch a graph of the foll…

00:44

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06:04

In this problem, we have been asked to find the inflection point of the function, which is given as why is equal to X cubed minus three X square. Now this is going to be obtained by determining the point export, which the second derivative of the function is equal to zero. So for that we need to determine the second derivative. First of all, that has determined the first derivative, that will be the derivative of X cubed, which is three X square minus three times the derivative of X square, which is two X. So this is equal to three X square minus six X. Next let us determine the second derivative that will be equal to the derivative of this expression over here. So that will be equal to three times the derivative of X square, which is two x minus six times the derivative of X, which is one. So this will be equal to six x minus six. Now, in order to find the point of inflection, we need to substitute the two ideas to that is the second derivative to be zero. So we obtain that zero is equal to six x minus six, hence six X will be equal to six and X will be six by six which is equal to one and at x equals to one. The value of Y is equal to x cube, so that's one q minus three X square. So this is three times one square. So this will be one minus three which is equal to minus two, so the value of X will be one and the value of Y will be minus two and hence the required inflection point will be the point 1 -2. Hence the required point is 1 -2.

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