   Chapter 10, Problem 13RE ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Find the relative maxima, relative minima, and points of inflection of the graph of. y = x 3 − 3 x 2 − 9 x + 10 .

To determine

To calculate: The relative minimum, relative maximum and points of inflection for the function y=x33x29x+10.

Explanation

Given Information:

The provided function is,

y=x33x29x+10

Formula used:

To find relative maxima and minima of a function,

1. Set the first derivative of the function to zero, f'(x)=0, to find the critical values of the function.

2. Substitute the critical values into f(x) and calculate the critical points.

3. Evaluate f(x) at each critical value for which f(x)=0.

(a) If f(x0)<0, a relative maximum occurs at x0.

(b) If f(x0)>0, a relative minimum occurs at x0.

(c) If f(x0)=0 or f(x0) is undefined, the second derivative test fails and then use the first derivative test.

Calculation:

Consider the provided function,

y=x33x29x+10

Now, calculate the first derivative.

y=x33x29x+10y=3x26x9=3(x22x3)=3(x3)(x+1)

Now, to obtain the critical values, set y=0 as,

3(x3)(x+1)=0

Thus, either x3=0 or x+1=0.

First consider x3=0.

x3+3=0+3x=3

And,

x+1=0x=1

Thus, the critical values of the function are at x=3 and x=1.

Now, substitute 3 for x in the equation y=x33x29x+10,

y=333(3)29(3)+10=17

Substitute 1 for x in the equation y=x33x29x+10,

y=(1)33(1)29(1)+10=15

Thus, the critical points are (3,17) and (1,15)

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