Determine Increasing/Decreasing and Concavity

Question

Given the function f(x)= 3x^4−12x^3, determine all intervals on which f is both decreasing and concave up.

Answer 1

f(x)= 3x^4−12x^3,,

f'(x)= 12x^3−36x^2,,

12x^3−36x^2=0

x=0, x=3

From this follows that x=0, x=3 are critical points

f''(x)=36x^2-72x (x=0 and x=2 are inclination points)

f''(x)=f''(2)=0,

f''(x)<0 at (0,2) and f''(x)>0 for x<0 and x>2,

Now can you see where the function is increasing or decreasing and where it is upward/downward?