Question

Which one of the following graphs describes the function f(x) = $e^{-x}(x^2+x+1)$ ?

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

$f\left(x\right)=e^{-x}(x^2+x+1)$

$f\left(x\right)>0\forall x$

$f\left(0\right)=1$

$f\left(\infty \right)=0$

$f^{\prime }\left(x\right)=e^{-x}(2x+1)-e^{-x}(x^2+x+1)$

$f^{\prime }\left(x\right)=e^{-x}(-x^2+x)$

Now, $f^{\prime }\left(x\right)=0\Rightarrow x=0,1$

$f^{\prime \prime }\left(x\right)=e^{-x}(-2x+1)-e^{-x}(-x^2+x)$

$=e^{-x}(x^2-3x+1)$

$f^{\prime \prime }\left(1\right)=e^{-1}(1-3+1)<0$

$\Rightarrow f\left(x\right)$ has local maxima at x = 1

$f^{\prime \prime }\left(0\right)=1>0$

$\Rightarrow f\left(x\right)$ has local minima at x = 0.
Among the given options, only option (b) satisfies these conditions.