Chapter 6.6, Problem 52STP

a.

To determine

To calculate: the x -coordinate of every turning point and also determine whether the coordinates are relative maximum or relative minima.

Answer to Problem 52STP

  $-2(\text{maximum}),3(\text{maximum}),0(\text{minimum}),3.5(\text{minimum})$ .

Explanation of Solution

Given information:

  

Formula used:

Relative maximum- The point on the graph is a relative maximum of the function when there is no other nearby points that have greater y -coordinate.

Relative minimum- The point on the graph is a relative minimum of the function when there is no other nearby points that have lesser y -coordinate.

Calculation:

According to the graph ,

We observed four turning points $-2,0,3\text{ and 3}.5$ at which the function changes.

At point $x=-2$ , the value of function is greater than the surrounding points, so there must be relative maximum.

At point $x=3$ , the value of function is greater than the surrounding points, so there must be relative maximum.

At point $x=0$ , the value of function is lesser than the surrounding points, so there must be relative minimum .

At point $x=3.5$ , the value of function is lesser than the surrounding points, so there must be relative minimum .

Hence, the following are the x -coordinate $-2(\text{maximum}),3(\text{maximum}),0(\text{minimum}),3.5(\text{minimum})$ .

b.

To determine

To calculate: the x -coordinate of every zero.

Answer to Problem 52STP

The x -coordinate of every zero are $x=-1,x=-3.8,x=1.5,x=3.5\text{ and }x=3.7$ .

Explanation of Solution

Given information:

  

Formula used:

The changes in sign indicate that there are zeros between those two x -values.

Calculations:

According to the graph ,

We observed four turning points $-2,0,3\text{ and 3}.5$ at which the function changes.

The graph crosses the x -axis at five points.

At $x=-1,x=-3.8,x=1.5,x=3.5\text{ and }x=3.7$ .

Hence, the x -coordinate of every zero are $x=-1,x=-3.8,x=1.5,x=3.5\text{ and }x=3.7$ .

c.

To determine

To calculate: the smallest possible degree of the function.

Answer to Problem 52STP

Smallest degree of the function is 5.

Explanation of Solution

Given information:

  

Formula used:

The maximum and minimum values of a function are called extrema .

These points are referred to as turning points.

The graph of the polynomial function of degree n has at most $n-1$ turning points.

Calculation:

According to the graph ,

We observed four turning points at which the function changes.

Hence, the smallest possible degree of the function is 5.

d.

To determine

To calculate: the domain and range of the function .

Answer to Problem 52STP

The domain and range of the function are all real numbers and $\left\{y|y\le 4\right\}\cup \left\{y|y\ge -2\right\}$ respectively.

Explanation of Solution

Given information:

  

Formula used:

Domain of the function : is the set of all values for which function is defined.

Range of the function : is the set of all values that f takes.

Calculation:

According to the graph ,

Domain is presented by x -axis. This implies domain is all real numbers.

Range is presented by y -axis. This implies $\left\{y|y\le 4\right\}\cup \left\{y|y\ge -2\right\}$ .

Hence, the domain and range of the function are all real numbers and $\left\{y|y\le 4\right\}\cup \left\{y|y\ge -2\right\}$ respectively.