Q. 23 - Precalculus Enhanced with Graphing Utilities

Question

Use a graphing utility to graph each function over the indicated interval. Approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing.  

$f\left(x\right)=2x^4-5x^3+2x+1 (-2,3)$

Step-by-Step Solution

Answer

The graph of the function over the given interval is given as

For the function, the local maximum value is $1.53$ and it occurs at $x=0.41$.

The local minimum values are $0.54$ and $-3.56$ and it occurs at $x=-0.34$ and $x=1.80$ respectively.

The graph is increasing over the interval $(-0.34,0.41)$ and $(1.80,3)$ and it is decreasing over the interval $(-2,-0.34)$ and $(0.41,1.80)$.

Step 1. Graph the function

Using a graphing utility the graph of the function $f\left(x\right)=2x^4-5x^3+2x+1$ over the interval $(-2,3)$ is given as 

Step 2. Local maximum and minimum value

The function has a local maximum at $0.41$, since for all $x$ close to $0.41$, we have $f\left(x\right)\le f(0.41)$. The local maximum value is $f(0.41)=1.53$.

The function has a local minimum at $x=-0.34$ and the local minimum value is $f(-0.34)=0.54$.

The function also has a local minimum at $x=1.80$ and the local minimum value is $f(1.8)=-3.56$. 

Step 3. Increasing and Decreasing interval

From the graph, it can be seen that the graph is decreasing in the intervals $(-2,-0.34)$ and $(0.41,1.80)$.

And the graph is increasing in the intervals $(-0.34,0.41)$ and $(1.80,3)$.