Is $f(x)=1-xe^(-3x)$ concave or convex at $x=-2$ ?

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Answer 1 · 2018-07-27T21:41:15

function is convex at $x=-2$

Given function:

$f(x)=1-xe^{-3x}$

$f'(x)=-x(-3e^{-3x})-e^{-3x}(1)$

$f'(x)=e^{-3x}(3x-1)$

$f''(x)=e^{-3x}(3)+(3x-1)(-3e^{-3x})$

$f''(x)=e^{-3x}(6-9x)$

$f''(-2)=e^{6}(6-9(-2))$

$=24e^{6}$

Since $f(-2)>0$ hence the given function is convex at $x=-2$

Is f(x)=1-xe^(-3x)f(x)=1-xe^(-3x) concave or convex at x=-2x=-2?