find the absolute maximum and minimum values of the function over the indicated interval. f(x)=2x^2 + 5 (a)…

Explanation:

Step1: Find the derivative

The derivative of $f(x)=2x^{2}+5$ is $f^\prime(x) = 4x$ using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$.

Step2: Find critical points

Set $f^\prime(x)=0$. So, $4x = 0$, which gives $x = 0$. But $x = 0$ is not in the interval $[3,6]$.

Step3: Evaluate the function at endpoints

Evaluate $f(x)$ at $x = 3$ and $x = 6$. When $x = 3$, $f(3)=2\times3^{2}+5=2\times9 + 5=18 + 5=23$. When $x = 6$, $f(6)=2\times6^{2}+5=2\times36+5 = 72 + 5=77$.

Answer:

The absolute maximum value is 77 at $x = 6$.