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.
$f(x)=\frac{3}{4}x^{4}+2x^{3}-4$
local extrema: (0 ×,4 ×)
increasing on the interval: (0,∞) ×
decreasing on the interval: (-∞,0) ×

Explanation:

Step1: Recall definition of local extrema

Local extrema occur where the function changes from increasing to decreasing (local maximum) or vice - versa (local minimum). From the graph, we can see a local minimum.

Step2: Identify local minimum

The function $f(x)=\frac{3}{4}x^{4}+2x^{3}-4$ has a local minimum at the point where the graph changes from decreasing to increasing. By observing the graph, the local minimum is at the point $(- 2,-8)$.

Step3: Recall definition of increasing and decreasing intervals

A function is increasing when the slope of the tangent line is positive and decreasing when the slope of the tangent line is negative.

Step4: Determine increasing interval

Looking at the graph, the function is increasing when $x>-2$. So the increasing interval is $(-2,\infty)$.

Step5: Determine decreasing interval

The function is decreasing when $x < - 2$. So the decreasing interval is $(-\infty,-2)$.

Answer:

Local extrema: $(-2,-8)$
Increasing on the interval: $(-2,\infty)$
Decreasing on the interval: $(-\infty,-2)$