Question

What are the critical values, if any, of `f(x)=x^(4/5) (x − 3)^2`?

Answer 1

They are `0`, `3`, and `6/7`.

Explanation:

`f(x)=x^(4/5) (x − 3)^2`

Critical value of `f` are values in the domain of `f` at which either `f'` does not exist or `f'(x)=0`

The domain of `f` is `RR`.

So every point at which `f'` fails to exist or `f'(x)=0` is a critical value for `f`.

Differentiate using the product rule:

`f'(x) = 4/5 x^(-1/5)(x-3)^2+x^(4/5) 2(x-3)(1)` `" "` (using the chain rule at the end)

`f'(x) = (4(x-3)^2)/(5root(5)x)+(2root(5)x^4(x-3))/1`

`= (4(x-3)^2+10x(x-3))/(5root(5)x)`

`= (2(x-3)[2(x-3)+5x])/(5root(5)x)`

`= (2(x-3)(7x-6))/(5root(5)x)`

`f'(0)` does not exist, but `0` is in the domain of `f`, so `0` is a critical value for `f`.

`f'(x) = 0` at `x=3` and at `x=6/7`, both of which are in the domain of `f`.

The critical values are `0`, `3`, and `6/7`.

We have finished the problem without looking at the graph of `f`. Having done that, it can be helpful to look at the graph.
You can zoom in and out and drag the graph around using a mouse. (It will start the same every time you return to this answer.)

graph{y=x^(4/5)(x-3)^2 [-3.01, 6.857, -0.442, 4.49]}