Question 1158620: Locate the critical points of the following function. Then use the Second Derivative Test to determine whether they correspond to local maxima, local minima, or neither.
f(x) = −e^x (x + 6)
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
I will presume that you have seen a proof that the product of two differentiable functions is, itself, differentiable, or if not, that you accept this fact on faith. Further, I assume that you accept the fact that \ =\ -e^x) and \ =\ (x\,+\,6)) are both differentiable on all reals.
Given the above, it is clear that there is no real number  such that ) does not exist. Hence, the only critical points are where \ =\ 0)
\ =\ \phi(x)\psi(x))
\ =\ \phi'(x)\psi(x)\ +\ \psi'(x)\phi(x))
\ =\ (-e^x)(x\,+\,6)\ +\ \(1)(-e^x)\ =\ -e^x(x\,+\,7))
And by inspection, we see that only \ =\ 0) , hence \)) is the only critical point of the function.
I leave it as an exercise for you to show that the second derivative is }(x)\ =\ -e^x(x\,+\,8)) , determine the sign of the value of }(-7)) , and use that result to select local maxima or minima based on:
}(x)\ <\ 0\ \Right\ ) Local maxima
}(x)\ >\ 0\ \Right\ ) Local minima
John
My calculator said it, I believe it, that settles it
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