Question

What are the local extrema, if any, of `f(x)= –2x^3 + 6x^2 + 18x –18`?

Answer 1

Maximum f is `f(5/2)` = 69.25. Minimum f is `f(-3/2)` = 11.25.

Explanation:

`d/dx(f(x))=-6x^2+12x+18 = 0`, when `x =5/2 and -3/2`

The second derivative is `-12x+12=12(1-x) < 0` at `x = 5/2` and > 0 at x = `3/2`.
So, f(`5/2`) is the local (for finite x) maximum and f(`-3/2`) is the local (for finite x) minimum.

As `xto oo, fto -oo` and as `xto-oo, fto+oo`..