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Find the critical numbers of the function.$f(x)=x^{3}+x^{2}-x$

$x=-1, x=\frac{1}{3}$

Calculus 1 / AB

Chapter 4

Applications of Differentiation

Section 1

Maximum and Minimum Values

Differentiation

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in this example, we have to find the critical values off if X were if X is given by Ex Cute plus X squared minus X now, to find the critical points, we have to sit the derivative off if X equal to zero. So to find the derivative, it will be three x squared plus two x minus one. Putting this equal to zero. We can find X, by fact arising, and it will be three weeks minus one and X plus one for the two factors, which gives us the values off eggs off one over three, the third or X equal to negative one. Right? So these this gives us the X values off the critical points, not necessarily the turning points that can only be proved by the second derivative, which is listen for another day to find the corresponding Why values? We have to substitute the values off this eggs. These two values off X into the original if off its in other words, the y value that corresponds with 1/3 real gate. We'll put the third into the original if off X and calculating this the Y value is negative. Five over 27 right? So, in other words, our co ordinates here is 1/3 and negative five over 27. The why value that corresponds off the negative one. It's found by plugging in the negative one into the original if off X, which will give us the value off, right. It gives us the value off unless one on this gives us the coordinates off negative one and one for the Wye Valley, Right? If you have a look at the graph, you can see on the growth that there are indeed turning points at point. D and E are using the second derivative taste. We can confirm that, but for now, this is all needed. The coordinates off the turning points off the critical points in this case.

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