Find the absolute maximum and minimum values of the function on each of the indicated intervals

Question

Find the absolute maximum and minimum values of the function on each of the indicated intervals.

 

f(x)=x^4-6x^2-1

 

a) Interval [-3,-1]

Absolute maximum:_______

Absolute minimum:_______

 

b) Interval [-4,1]

Absolute maximum:_______

Absolute minimum:_______

 

c) Interval [-3.4]

Absolute maximum:_______

Absolute minimum:_______

Answer 1

The function does not have a absolute maximum.

A relative maximum is at x = 0, f(0) = -1

 

The relative minimums are at x = ±√3, where f(±√3) = -10

Answer 2

I'll do part (a).  The other parts are done in a similar manner.  

 

Since f(x) is a polynomial function, it is continuous on any interval.

 

The absolute max and min must occur at a critical point in the interval or at an endpoint.  

 

f'(x) = 4x³ - 12x = 4x(x² - 3) = 0 when x = 0, ±√3

 

Critical points in the  interval [-3,-1]:  0, -√3

 

f(0) = -1

 

f(-√3) = -10

 

f(-3) = 26

 

f(-1) = -6

 

absolute max of f on [-3,-1]:  26

 

absolute min on [-3,-1]:  -10