find the critical points of the following function on the given interval. identify the absolute maximum and…

find the critical points of the following function on the given interval. identify the absolute maximum and absolute minimum values (if they exist). f(x)=5x³ - 30x² on -1,5 identify all the critical points on the given interval. select the correct choice below and, if necessary, fill in the answer box within your choice. a. the critical point(s) occur(s) at x = (use a comma to separate answers as needed.) b. there are no critical points for f.

find the critical points of the following function on the given interval. identify the absolute maximum and absolute minimum values (if they exist). f(x)=5x³ - 30x² on -1,5 identify all the critical points on the given interval. select the correct choice below and, if necessary, fill in the answer box within your choice. a. the critical point(s) occur(s) at x = (use a comma to separate answers as needed.) b. there are no critical points for f.

Answer

Explanation:

Step1: Find the derivative

$f'(x)=\frac{d}{dx}(5x^{3}-30x^{2}) = 15x^{2}-60x$

Step2: Set the derivative equal to 0

$15x^{2}-60x = 0$ Factor out $15x$: $15x(x - 4)=0$ Solve for $x$: $x = 0$ or $x = 4$ Both $0$ and $4$ are in the domain of the original function. And $0\in[-1,5]$, $4\in[-1,5]$.

Step3: Evaluate the function at critical points and endpoints

Evaluate $f(x)$ at $x=-1$: $f(-1)=5\times(-1)^{3}-30\times(-1)^{2}=5\times(-1)-30\times1=-5 - 30=-35$ Evaluate $f(x)$ at $x = 0$: $f(0)=5\times0^{3}-30\times0^{2}=0$ Evaluate $f(x)$ at $x = 4$: $f(4)=5\times4^{3}-30\times4^{2}=5\times64-30\times16=320 - 480=-160$ Evaluate $f(x)$ at $x = 5$: $f(5)=5\times5^{3}-30\times5^{2}=5\times125-30\times25=625 - 750=-125$

Answer:

A. The critical point(s) occur(s) at $x = 0,4$ The absolute maximum value is $0$ (at $x = 0$) and the absolute minimum value is $-160$ (at $x = 4$)