What are the critical points of $f (x) = x \sqrt{e^{x} - 3 x}$ $f(x) =xsqrt(e^x-3x)$ ?

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Answer 1 · 2016-02-08T18:12:07

They are approximately $0.619$ $0.619$ , $1.512$ $1.512$ , and $0.395$ $0.395$ .

$f'(x) = (xe^x-2e^x-9x)/(2sqrt(e^x-3x)$

The critical numbers of $f$ are the solutions to

$e^x-3x=0$

and the solution to $xe^x-2e^x-9x=0$ that gives $e^x-3x >= 0$ (so that it is in the domain of $f$ .

Use whatever numerical/technological methods you have to get approximations.

$0.619$ and $1.512$ solve the first equation and

$0.395$ and $1.234$ solve the second, but $e^1.234-3(1.234) < 0$ so $1.234$ is not in the domain of $f$ and hence is not a critical number for $f$ .

What are the critical points of f(x) =xsqrt(e^x-3x)f(x)=x√ex−3xf(x) =xsqrt(e^x-3x)?