Question

how to find the maximum and minimum?

Consider the function: `f R ->R` given `f(x)= 16x-x^3` .

Determine the local maximum / minimum points `f` and the values ​​of `f`.

Answer 1

Please see below.

Explanation:

Find the critical numbers.
These are numbers in the domain of `f` at which the derivative is either `0` or fails to exist.

`f(x)=16x-x^3` has domain `(-oo,oo)`.

`f'(x) = 16-3x^2`

`f'(x)` exists for all `x`, and is `0` at

`16-3x^2=0`

`x^2=16/3`

`x = +- 4/sqrt3 = +- (4sqrt3)/3`
These are the critical numbers for `f`.

Test the critical numbers

Investigate the sign of `f'(x)`

On `(-oo,-(4sqrt3)/3)`, `f'(x) < 0` so `f` is decreasing.

On `(-(4sqrt3)/3, (4sqrt3)/3)`, `f'(x) > 0` so `f` is increasing.

Therefore `f(-(4sqrt3)/3)` is a local minimum value of `f`. (The arithmetic is left to the reader.)

On `(-(4sqrt3)/3, (4sqrt3)/3)`, `f'(x) > 0` so `f` is increasing.

On `((4sqrt3)/3,oo)`, `f'(x) <0` so `f` is decreasing.

Therefore `f((4sqrt3)/3)` is a local maximum value of `f`. (The arithmetic is left to the reader.)

I'm not sure what you have been taught to call maximum points, but I hope this is enough for you to find them.