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Numerade Educator

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Problem 40 Medium Difficulty

(a) If $ f(x) = (x^2 - 1) e^x, $ find $ f' (x) $ and $ f" (x). $
(b) Check to see that your answers to part (a) are reasonable by comparing the graphs of $ f, f', $ and $ f". $

Answer

(a) $f^{\prime}(x)=e^{x}\left(x^{2}+2 x-1\right)$ and $f^{\prime \prime}(x)=e^{x}\left(x^{2}+4 x+1\right)$
(b) -14

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Video Transcript

Yes. Close. So when you married here, so we have ever backs is equal to x square minus one times e to the X. We're gonna find our derivative by using the product rule yet to ex homes, Eat the X goes eat two x times X square minus one. This is equal to eat the X x square plus to minus one. To get our second derivative, we're gonna use the product rule again. Get E to the X from specs square Those two X minus one must two X plus two e to the X, which is equal to eat The axe. Times Square was for X plus one. Next, we're going to draw our graphs wound. Don't look like this.