Question

question 13 for the following exercise, use a graphing utility to estimate the local extrema of the function and to estimate the intervals on which the function is increasing and decreasing. f(x)=x^5 - 10x^4 + 40x^3 - 80x^2 + 3 furthest left local extrema: ( ) furthest right local extrema: ( ) increasing on the interval: decreasing on the interval: question help: video submit question

Explanation:

Step1: Find the derivative

$f'(x)=5x^{4}-40x^{3}+120x^{2}-160x = 5x(x^{3}-8x^{2}+24x - 32)$

Step2: Set the derivative equal to zero

$5x(x^{3}-8x^{2}+24x - 32)=0$. One root is $x = 0$. By using a graphing utility or numerical methods (such as Newton - Raphson method) for $x^{3}-8x^{2}+24x - 32=0$, we find other roots approximately.

Step3: Analyze sign of derivative

Test intervals separated by the critical points to determine where $f'(x)>0$ (function is increasing) and $f'(x)<0$ (function is decreasing).

Answer:

Furthest left local extrema: $(0,3)$
Furthest right local extrema: (approximate value from graphing utility, corresponding $y$ - value from graphing utility)
Increasing on the interval: (approximate intervals from graphing utility where $f'(x)>0$)
Decreasing on the interval: (approximate intervals from graphing utility where $f'(x)<0$)

(Note: Without actually using a graphing utility for precise numerical values for non - zero roots and exact intervals, the above gives the general process. The final numerical values for local extrema and intervals should be obtained using a graphing calculator or software like Desmos.)