Question

If $g\left(x\right)=7x^2{e}^{-x^2}\forall x\in R,$ then g(x) has

• A. local maximum at $x=0$
• B. local minima at $x=0$
• C. local maximum at $x=1$
• D. two local maxima and one local minima

Answer 1

Answer: (B), (C), (D)

local minima at $x=0$
local maximum at $x=1$
two local maxima and one local minima

$g\left(x\right)=7x^2{e}^{-x^2}\forall x\in R,$
$g^{\prime }\left(x\right)=7(e^{-x^2}.2x-e^{-x^2}.2x.x^2)=14e^{-x^2}(x-x^3)$
$g^{\prime \prime }\left(x\right)=14e^{-x^2}(1-5x^2+2x^4)$
Now for maxima or minima of $g\left(x\right)$
$g^{\prime }\left(x\right)=0\Rightarrow x=0,-1,1$
Clearly $g^{\prime \prime }\left(1\right)<0,g^{\prime \prime }(-1)<0,g^{\prime \prime }\left(0\right)>0$
Hence $x=1,-1$ are local maxima point and $x=0$ is minima point.