Q. 6 - Precalculus Enhanced with Graphing Utilities

Question

Use a graphing utility to graph the function $f\left(x\right)=-x^4+2x^3+4x^2-2$ on the interval $(-5,5)$. Approximate any local maximum values and local minimum values rounded to two decimal places. Determine where the function is increasing and where it is decreasing.

Step-by-Step Solution

Answer

The graph is

$f\left(x\right)$ has a local maximum of $-0.86$ at $x=-0.85$ and $15.55$ at $x=2.35$.

$f\left(x\right)$ has a local minimum of $-2$ at $x=0$.

$f\left(x\right)$ is increasing on the interval $(-5,-0.85)$ and $(0,2.35)$.

$f\left(x\right)$ is decreasing on the interval $(-0.85,0)$ and $(2.35,5)$

Step 1. Draw the graph of the function.

The graph is

Step 2. Find the local maximum values and local minimum values. Also find where the function is increasing and where it is decreasing.

From the graph, we see 

$f\left(x\right)$ has a local maximum of $-0.86$ at $x=-0.85$ and $15.55$ at $x=2.35$.

$f\left(x\right)$ has a local minimum of $-2$ at $x=0$.

$f\left(x\right)$ is increasing on the interval $(-5,-0.85)$ and $(0,2.35)$.

$f\left(x\right)$ is decreasing on the interval $(-0.85,0)$ and $(2.35,5)$