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6th Edition

Stewart + 5 others

Publisher: Cengage Learning

ISBN: 9780840068071

Chapter 3.2, Problem 4E

(a)

To determine

**To check:** The given statement is true or not.

Expert Solution

The given statement is not true.

**Given:**

The polynomial *P* has 3 degree. The given polynomial has two local maxima and two local minima.

**Calculation:**

Let us consider a polynomial *P* of degree n.

Then the graph of *P* has at most

Local extrema is the points on the graph that has maximum and minimum value.

If a polynomial has 3 degree then the no of extrema is 2.

The given statement is that polynomial has 2 local maxima and 2 local minima that means there are 4 local extrema, but the polynomial have only 2 extrema.

Thus, the given statement is not true.

(b)

To determine

**To check:** The given statement is true or not.

Expert Solution

The given statement is true.

**Given:**

The polynomial *P* has 3 degree. The given polynomial has no local maxima or minima.

**Calculation:**

Let us consider a polynomial *P* of degree n.

Then the graph of *P* has at most

Local extrema is the points on the graph that has maximum and minimum value.

If a polynomial has 3 degree then at most 2 local extrema, but there is no minimum number for local minima and maxima.

So, it may be possible that there should not be any local maxima or minima.

Thus, the given statement is true.

(c)

To determine

**To check:** The given statement is true or not.

Expert Solution

The given statement is true.

**Given:**

The polynomial *P* has 4 degree. The given polynomial has one local maximum and no minima.

**Calculation:**

Let us consider a polynomial *P* of degree n.

Then the graph of *P* has at most

Local extrema is the points on the graph that has maximum and minimum value.

If a polynomial has 4 degree then at most 3 local extrema, but there is no minimum number for local minima and maxima.

So, it may be possible that there should one local maximum and no local minima.

Thus, the given statement is true.