# Local maximum and local minimum values of function and value of x at which they occur. ### 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 2, Problem 72RE
To determine

## To find: Local maximum and local minimum values of function and value of x at which they occur.

Expert Solution

Local minimum at x=0 and no local maxima.

### Explanation of Solution

Given information:

f(x)=x23(6x)13

Calculation:

For find local maxima or minima first find critical point of function.

For critical point of function f'(x)=0

f'(x)= d dx ( x 2 3 (6x) 1 3 ) = (6x) 1 3 × d dx ( x 2 3 )+ x 2 3 × d dx (6x) 1 3 = (6x) 1 3 × 2 3 (x) 1 3 + x 2 3 × 1 3 × (6x) 2 3 ×1 = 1 3 x 1 3 (6x) 2 3 ( 2(6x)+ x 1 3 )

For critical point

13x13(6x)23(2(6x)+x13)=0

From this only x13=0 or x=0

Hence function has one critical point x=0

Now sign of f’(x) on both side of x=0

So value of derivative at x=1

f'(1)= 1 3 (1) 1 3 (6(1)) 2 3 (2(6(1))+ (1) 1 3 ) = 1 3 ×1× (7) 2 3 (141) = 13 3 (7) 2 3 <0

Value of derivative at x=1

f'(1)= 1 3 (1) 1 3 (6(1)) 2 3 (2(6(1))+ (1) 1 3 ) = 1 3 ×1× (5) 2 3 (10+1) = 11 3 (5) 2 3 >0

So f’(x) changing sign from negative to positive. So there is a local minimum at x=0 and no local maxima for function.

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