use the graph below to estimate the local extrema of the function and to estimate the intervals on which the…

use the graph below to estimate the local extrema of the function and to estimate the intervals on which the function is increasing and decreasing.\n$f(x)=x^{5}-5x^{4}+10x^{3}-10x^{2}-2$\nfurthest left local extrema: \nfurthest right local extrema: -10 \nincreasing on the interval: \ndecreasing on the interval:

use the graph below to estimate the local extrema of the function and to estimate the intervals on which the function is increasing and decreasing.\n$f(x)=x^{5}-5x^{4}+10x^{3}-10x^{2}-2$\nfurthest left local extrema: \nfurthest right local extrema: -10 \nincreasing on the interval: \ndecreasing on the interval:

Answer

Explanation:

Step1: Recall local extrema concept

Local extrema are points where the function changes from increasing to decreasing (local maximum) or vice - versa (local minimum).

Step2: Identify furthest left local extrema from graph

By observing the graph, the furthest left local extrema occurs around $x=- 1$. Substitute $x = - 1$ into $f(x)=x^{5}-5x^{4}+10x^{3}-10x^{2}-2$: $f(-1)=(-1)^{5}-5(-1)^{4}+10(-1)^{3}-10(-1)^{2}-2=-1 - 5-10 - 10-2=-28$. So the furthest left local extrema is $(-1,-28)$.

Step3: Identify furthest right local extrema from graph

From the graph, the furthest right local extrema is given as $(3,-10)$.

Step4: Determine increasing intervals

A function is increasing when the slope of the tangent line is positive. From the graph, the function $f(x)$ is increasing on the intervals $(-\infty,-1)$ and $(1,3)\cup(3,\infty)$.

Step5: Determine decreasing intervals

A function is decreasing when the slope of the tangent line is negative. From the graph, the function $f(x)$ is decreasing on the intervals $(-1,1)$.

Answer:

Furthest left local extrema: $(-1,-28)$ Furthest right local extrema: $(3,-10)$ Increasing on the interval: $(-\infty,-1),(1,3)\cup(3,\infty)$ Decreasing on the interval: $(-1,1)$