Chapter 4.8, Problem 31E

To determine

To estimate: the coordinates of given turning point, to state whether h=given corresponds to a local maximum or a local minimum, and to estimate the real zeros and to find the least possible degree of the function.

Answer to Problem 31E

Least degree of the function is 3.

The local maximum is at about $\left(-0.25,0.5\right)$ .

The local minimum is at about $\left(0.25,-0.5\right)$ .

The real zeros are $-0.5,0$ and 0.5.

Explanation of Solution

Given information:

Given graph,

  

Calculation:

From the graph it is very clear that it has two turning points.

Because it has two points, so least degree of the function is $2+1=3$ .

The $y$ coordinate of a turning is a local maximum of the function when the point is higher than all nearby points.

From the graph it is very clear that, the local maximum is at about $\left(-0.25,0.5\right)$ .

The $y$ coordinate of a turning is a local minimum of the function when the point is higher than all nearby points.

From the graph it is very clear that, the local minimum is at about $\left(0.25,-0.5\right)$ .

From the graph it is very clear that, it crosses $x$ axis at about $x=-0.5,x=0$ and $x=0.5$ .

Therefore, the real zeros are $-0.5,0$ and 0.5.