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QUESTION

Find and classify the stationary point of the function
$f(x)=xe^x$.

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ANSWER

$\ds f(x)=xe^x,\ \frac{df}{dx}=xe^x+e^x=(x+1)e^x,\ =0$ for
stationary point. $e^x\neq0,$ therefore $x+1=0,\ x=-1$ is the
stationary point.

$\ds\frac{d^2f}{dx^2}=(x+1)e^x+e^x=(x+2)e^x$

When $\ds x=-1,\ \frac{d^2f}{dx^2}=(-1+2)e^{-1}=\frac{1}{e}>0.$

Therefore the stationary point is a minimum.




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