   Chapter 10, Problem 14RE ### 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

# In Problems 14 and 15, find any relative maxima, relative minima, and points of inflection, and sketch each graph. y = x 3 − 12

To determine

To calculate: The relative minimum, relative maximum and points of inflection for the function y=x312x and sketch its graph.

Explanation

Given Information:

The provided function is,

y=x312x

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=x312x

Now, calculate the first derivative.

y=x312xy=3x212=3(x24)=3(x2)(x+2)

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

3(x2)(x+2)=0

Thus, either x2=0 or x+2=0.

First consider x2=0.

x2+2=0+2x=2

And,

x+2=0x+22=02x=2

Thus, the critical values of the function are at x=2 and x=2.

Now, substitute 2 for x in the equation y=x312x,

y=2312(2)=16

Now, substitute 2 for x in the equation y=x312x,

y=(2)312(2)=16

Thus, the critical points are (2,16) and (2,16)

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