this is similar to section 3.7 problem 12: for the function f(x)=5x^2 + 3x+( - 1), determine the absolute…

Explanation:

Step1: Find the derivative

The derivative of $f(x)=5x^{2}+3x - 1$ is $f'(x)=10x + 3$.

Step2: Find critical points

Set $f'(x)=0$, so $10x+3 = 0$, then $x=-\frac{3}{10}$. But $-\frac{3}{10}\notin[0,4]$, so we ignore it.

Step3: Evaluate the function at endpoints

Evaluate $f(x)$ at $x = 0$ and $x = 4$. When $x = 0$, $f(0)=5\times0^{2}+3\times0 - 1=-1$. When $x = 4$, $f(4)=5\times4^{2}+3\times4-1=5\times16 + 12-1=80 + 12-1=91$.

Answer:

Absolute maximum = 91 at x = 4. Absolute minimum = -1 at x = 0.