find the absolute maximum and minimum values of the following function on the given interval. identify the…
find the absolute maximum and minimum values of the following function on the given interval. identify the points on the graph where the absolute extrema occur.\nf(x)=-\frac{6}{x^{2}},1leq xleq2\nfind the absolute maximum. select the correct choice below and, if necessary, fill in the answer choice.\na. the absolute maximum value occurs at x = \n(use a comma to separate answers as needed.)\nb. there is no absolute maximum.
Answer
Explanation:
Step1: Find the derivative
The derivative of $f(x)=-\frac{6}{x^{2}}=-6x^{-2}$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $f^\prime(x)=12x^{-3}=\frac{12}{x^{3}}$.
Step2: Find critical points
Set $f^\prime(x) = 0$. Since $\frac{12}{x^{3}}\neq0$ for any real $x\neq0$, there are no critical points in the domain of $f(x)$ where the derivative is zero. Also, the function $f(x)=-\frac{6}{x^{2}}$ is continuous on the closed interval $[1,2]$.
Step3: Evaluate the function at endpoints
Evaluate $f(x)$ at $x = 1$ and $x = 2$. When $x = 1$, $f(1)=-\frac{6}{1^{2}}=-6$. When $x = 2$, $f(2)=-\frac{6}{2^{2}}=-\frac{6}{4}=-\frac{3}{2}$.
Answer:
A. The absolute maximum value $-\frac{3}{2}$ occurs at $x = 2$.