Question

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.
$f(x)=x^{5}-5x^{4}+10x^{3}-10x^{2}-2$
furthest left local extrema:
furthest right local extrema: -10
increasing on the interval:
decreasing on the interval:

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: $(-1,1)$