Question

How do you find the critical points `f(x)= x^3 + 35x^2 - 125x - 9375`?

Answer 1

The critical numbers are: `x = -25` and `x=5/3`

Explanation:

A critical number for `f` is a number in the domain of `f` at which, `f' =0` or `f'` does not exist.

`f(x)= x^3 + 35x^2 - 125x - 9375`

`f'(x) = 3x^2+70x-125`

Clearly, this function exists for all `x`, so we need only consider its zeros:

`3x^2+70x-125 = 0`

To look for factors using whole numbers:
`3 xx -125= - 375`
Find two numbers whose product is `-375` and whose sum is `70`

`-1 xx 375` won't work
`-2` is not a factor of 375
`-3 xx 125` won't work
`-5 xx 75` STOP! that's the one.
Now split the middle term and factor by grouping:

`3x^2+70x-125 = 0`

`3x^2-5x+75x-125 = 0`

`x(3x-5)+25(3x-5) = 0`

`(x+25)(3x-5) = 0`

`x = -25` and `x=5/3`. Both are in the domain of `f` (Domain of `f = RR`), so they are the critical numbers for `f`.