Find the extreme values of the function and where they occur.
-$$y = x ^ { 2 } e ^ { x }$$
A) Minimum value is 0 at $x = 0$; no maximum value.
B) Minimum value is 0 at $x = 0$, maximum value is $4 e ^ { - 2 }$ at $x = - 2$.
C) Minimum value is $4 e ^ { - 2 }$ at $x = - 2$; no maximum value.
D) None

Question

Quizplus · Accepted Answer

To find the extreme values of the function $y = x^2e^x$, we first find its derivative: $$y' = 2xe^x + x^2e^x = x e^x (2 + x)$$. Setting $y' = 0$ gives $x = 0$ or $x = -2$. At $x = 0$, the function has a minimum value since the second derivative test ($y'' > 0$) confirms it's a point of concavity upwards. At $x = -2$, the function value is $4e^{-2}$, but this is not a maximum since the function grows without bound for $x > 0$. Therefore, the minimum value is 0 at $x = 0$, and there is no maximum value.

Find the Extreme Values of the Function and Where They y=x2exy = x ^ { 2 } e ^ { x }y=x2ex

Find the Extreme Values of the Function and Where They $y = x ^ { 2 } e ^ { x }$