Question

Find the absolute (or global) maximum and absolute (or global) minimum values of the function `f(x)=x+4/x^2` on the interval [1,4]?

`f(x)=x+4/x^2`

Answer 1

Start by finding the derivative.

`f'(x) = 1 - 8/x^3`

This has critical points at

`0 = 1- 8/x^3`

`8/x^3 = 1`

`x^3 = 8`

`x = 2`

This will either be a maximum or a minimum. At `x = 1`, the derivative has a negative value, thus this will be a minimum. The last step is to check the endpoints:

`f(1) = 1 + 4/1^2 = 5`
`f(2) = 2 + 4/2^2 = 3`
`f(4) = 4 + 4/4^2 = 17/4`

Thus the local maximum (we're only considering your given domain) will be `(1, 5)` and the local minimum will be `(2, 3)`.

Hopefully this helps!