Question

What are the absolute extrema of `f(x)=9x^(1/3)-3x in[0,5]`?

Answer 1

The absolute maximum of `f(x)` is `f(1)=6` and the absolute minimum is `f(0)=0`.

Explanation:

To find the absolute extrema of a function, we need to find its critical points. These are the points of a function where its derivative is either zero or does not exist.

The derivative of the function is `f'(x)=3x^(-2/3)-3`. This function (the derivative) exists everywhere. Let's find where it is zero:

`0=3x^(-2/3)-3rarr3=3x^(-2/3)rarrx^(-2/3)=1rarrx=1`

We also have to consider the endpoints of the function when looking for absolute extrema: so the three possibilities for extrema are `f(1), f(0)` and `f(5)`. Calculating these, we find that `f(1)=6, f(0)=0,` and `f(5)=9root(3)(5)-15~~0.3`, so `f(0)=0` is the minimum and `f(1)=6` is the max.