Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme
values, if any, are absolute.
-$h ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 8 x + 2 , \Leftrightarrow

Question

Identify the function's extreme values in the given domain, and say where they are assumed. Tell which of the extreme
values, if any, are absolute.
-$h ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 8 x + 2 , \Leftrightarrow < x \leq 0$
A) local and absolute maximum: 2 at x = 0;
B) no local extrema; no absolute extrema
C) local and absolute maximum: 2 at x = 0;.
D) local and absolute minimum: 1 at x = -1; local and absolute minimum: 1 at x = -1

Quizplus · Accepted Answer

The function $h(x) = x^3 + 3x^2 + 8x + 2$ is a cubic polynomial, which is continuous and differentiable everywhere. To find the extreme values, we first find the derivative: $h'(x) = 3x^2 + 6x + 8$. Setting $h'(x) = 0$ to find critical points gives a quadratic equation, but since the domain is restricted to $x \leq 0$, we only consider values within this domain. At the boundary of the domain, $x = 0$, $h(0) = 2$, which is the highest value the function attains in the given domain, making it both a local and absolute maximum. There are no other extrema within the domain that satisfy the conditions better than $x = 0$.

Identify the Function's Extreme Values in the Given Domain, and Say