Question

How do you find the local max and min for `f(x) = x^4 - 2x^2 + 1`?

Answer 1

max at (0 ,1), mins at (-1 ,0) and (1 ,0)

Explanation:

Differentiate f(x) and equate to zero to find `color(blue)"critical points"`

`rArrf'(x)=4x^3-4x`

`4x^3-4x=0rArr4x(x^2-1)=0rArr4x(x-1)(x+1)=0`

`rArrx=0,x=-1,x=1`

Find the y coordinates.

`f(0)=0-0+1=1rArr(0,1)`

`f(-1)=1-2+1=0rArr(-1,0)`

`f(1)=1-2+1=0rArr(1,0)`

To find if max/min use the `color(blue)"second derivative test"`

• If f''(a) > 0 , then minimum

• If f''(a) < 0 , then maximum

The second derivative is.

`f''(x)=12x^2-4`

and `f''(0)=-4<0rArr (0,1)" is a maximum"`

`f''(-1)=8>0rArr(-1,0)" is a minimum"`

`f''(1)=8>0rArr(1,0)" is a minimum"`
graph{x^4-2x^2+1 [-10, 10, -5, 5]}