   Chapter 10.3, Problem 2E ### 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 1-6, find the absolute maxima and minima for f ( x ) on the interval [ a ,   b ] . 2. f ( x ) = x 3 − 3 x + 3 , [ − 3 , 1.5 ]

To determine

To calculate: The absolute maxima and minima for f(x)=x33x+3 in the interval [3,1.5].

Explanation

Given Information:

The provided function is:

f(x)=x33x+3

Formula used:

If f(x) and g(x) are two differentiable functions then by the property of derivative:

ddx(f(x)+g(x))=ddxf(x)+ddxg(x)

And

ddxxn=nxn1

Where n is a constant and x is the variable.

Calculation:

The absolute maxima and absolute minima will occur only at the critical points. To calculate the critical points of the provided function find the first derivative of the function:

f(x)=x33x+3f(x)=ddx(x33x+3)

Use ddx(f(x)+g(x))=ddxf(x)+ddxg(x),

f(x)=ddxx3ddx3x+ddx3

Now use ddxxn=nxn1

f(x)=ddxx3ddx3x+ddx3=3x23+0=3x23

Now equate it to zero and find out the critical points of the provided function,

3x23=03(x1)(x+1)=0

So, x=1,1

Therefore, the critical points are 1,1

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