Question

What are the critical points of `f(x) =xsqrt(e^x-3x)`?

Answer 1

They are approximately `0.619`, `1.512`, and `0.395`.

Explanation:

`f'(x) = (xe^x-2e^x-9x)/(2sqrt(e^x-3x)`

The critical numbers of `f` are the solutions to

`e^x-3x=0`

and the solution to `xe^x-2e^x-9x=0` that gives `e^x-3x >= 0` (so that it is in the domain of `f`.

Use whatever numerical/technological methods you have to get approximations.

`0.619` and `1.512` solve the first equation and

`0.395` and `1.234` solve the second, but `e^1.234-3(1.234) < 0` so `1.234` is not in the domain of `f` and hence is not a critical number for `f`.