a. find the critical points of f on the given interval. b. determine the absolute extreme values of f on the…

Explanation:

Step1: Find the derivative of $f(x)$

Use the quotient - rule. If $f(x)=\frac{u}{v}$ where $u = 5x$ and $v=(x - 8)^{\frac{1}{2}}$, then $u'=5$ and $v'=\frac{1}{2}(x - 8)^{-\frac{1}{2}}$. The quotient - rule states that $f'(x)=\frac{u'v - uv'}{v^{2}}$. [ \begin{align*} f'(x)&=\frac{5\sqrt{x - 8}-5x\cdot\frac{1}{2\sqrt{x - 8}}}{x - 8}\ &=\frac{\frac{10(x - 8)-5x}{2\sqrt{x - 8}}}{x - 8}\ &=\frac{10x-80 - 5x}{2(x - 8)^{\frac{3}{2}}}\ &=\frac{5x-80}{2(x - 8)^{\frac{3}{2}}} \end{align*} ]

Step2: Find the critical points

Set $f'(x)=0$ and solve for $x$. [ \begin{align*} \frac{5x-80}{2(x - 8)^{\frac{3}{2}}}&=0\ 5x-80&=0\ 5x&=80\ x& = 16 \end{align*} ] Also, check where $f'(x)$ is undefined. The derivative $f'(x)$ is undefined when $x=8$, but $x = 8$ is not in the interval $[10,40]$.

Answer:

A. The critical point(s) occur(s) at $x = 16$