# The maximum value of the function f ( x ) = 3 + 4 x 2 − x 4 .

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 3.1, Problem 80E
To determine

## To find: The maximum value of the function f(x)=3+4x2−x4 .

Expert Solution

The maximum value of the function f(x)=3+4x2x4 is equal to 7 .

### Explanation of Solution

Given information:

The function is f(x)=3+4x2x4 .

Calculation:

Substitute t for x2 in the given function.

f(x)=3+4x2x4f(x)=3+4x2(x2)2f(t)=3+4tt2

Compare the quadratic function in t with at2+bt+c . This gives a=1 , b=4 and c=3 .

The maximum value of the function in t occurs at t=b2a i.e. t=42=2 .

So, the maximum value of the function f(t)=3+4tt2 occurs at t=2 .

Substitute 2 for t in the function.

f(t)=3+4tt2f(2)=3+4(2)(2)2f(2)=3+84f(2)=7

The maximum value of the function f(t)=3+4tt2 is equal to 7 and also the maximum value of the function f(x)=3+4x2x4 is equal to 7 at x=2=±1.414 .

Therefore, the maximum value of the function f(x)=3+4x2x4 is equal to 7 .

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