findthe maximum function: f(X)=1-x^4 using the derivative

The maximum and minimum values of a function are found where the first derivatives are equal to zero.

First, find the first derivative of f(x) = 1 - x^4

     --->  f'(x)  = -4x^3

Then, set f'(x) = 0

    --->  -4x^3  = 0  ---> x = 0

This says that any maximum or minimum occurs at the point  (0,y)

At x = 0  ---> f(0)  =  1 - 0^4   --->  f(0)  = 1

Thus, the maximum or minimum occurs at  (0, 1).

To see if it is a max or a min:

Test to the left of the point (0, 1):  f'(-1)  =  -4(-1)^3  =  4  (since positive, the slope indicates that the graph is rising to the left of the point.

Test to the right of the point (0, 1): f'(1)  =  -4(1)^3  =  -4 (since negative, the slope indicates that the graph is falling to the right of the point.

Thus, the point  (0, 1) is a maximum.