Question

Consider the function f(x)= (3/4)x^4-x^3-3x^2+6x

Find the relative extrema for f(x); be sure to label each as a maximum or minimum. You do not need to find function values; just find the x-values.
Determine the interval(s) where f(x) is increasing (if any) and the interval(s) where f(x) is decreasing (if any). Show why there are exactly two inflection points for this function. Note: You do NOT need to find the inflection points.

Any help on this would be much appreciated!

Answer 1

f = (3/4)x^4-x^3-3x^2+6x
f' = 3x^3-3x^2-6x+6
f" = 9x^2-6x-6

extrema where f'=0 and f"≠0
f increasing where f' > 0

there are two inflection points because f" has two real roots.