Find if any critical numbers of function f(x)=x^5-10x^3?

To find critical numbers, you can look for points where
the first derivative is undefined or zero. So to start, let's find the first
derivative:

`f(x) = x^5 - 10x^3 implies f'(x) = 5x^4 -
30x^2 = 5x^2(x^2 - 6)`

Next we must find the
zeros:

`5x^2(x^2 - 6) = 0 implies x = 0, x = sqrt(6), x =
-sqrt(6)`

We have now found the critical points and we see
that there are three of them. To find the critcal numbers, we must evaulate f(x) at each
of there points:

`c_1 = f(0) = 0^5 - 10(0)^3 =
0`

`c_2 = f(sqrt{6}) = (sqrt{6})^5 - 10(sqrt{6})^3 =
36sqrt{6} - 60sqrt{6} = -24sqrt{6}`

` ` `c_3 = f(-sqrt{6})
= (-sqrt{6})^5 - 10(-sqrt{6})^3 = -36sqrt{6} + 60sqrt{6} =
24sqrt{6}`