Chapter 4, Problem 50RE

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Using the Second Derivative Test In Exercises 49-56, find all relative extrema of the function. Use the Second Derivative Test where applicable. f ( x ) = x 4 − 2 x 2 + 6

To determine

To calculate: The relative extrema of the function f(x)=x42x2+6 using the second derivative test.

Explanation

Given:

The function f(x)=x4âˆ’2x2+6.

Formula Used:

For a function f that is twice differentiable on an open interval I, if f'(c)=0 for some c, then,

If f''(c)>0 the function f has relative minima at c if f''(c)<0 the function f has relative maxima at c.

Calculation:

First differentiate the provided function,

f'(x)=4x3âˆ’4x

Equate the first derivative to zero to obtain the critical points.

4x3âˆ’4x=04x(x2âˆ’1)=0x=âˆ’1,0,1

Now differentiate the function for the second tine and replace this for x:

f''(âˆ’1)=12(âˆ’1)2âˆ’4=12âˆ’4=8>0

This implies that the function has relative minima at this point.

And,

f''(0)=12(0)2âˆ’4=âˆ’4<0

This implies that the function has relative maxima at this point

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