Question

The function defined by f(x)=x^3-3x^2 for all real numbers x has a relative maximum at x= ?

Answer 1

max and mins occur at f'(x)=0

f(x)=x^3-3x^2
f'(x)=3x^2-6x=3x(x-2)=0--->x={0,2}

to find if this is a min or max we must look at f''(x)

f''(x)=6x-6=0--->x=1
so for x<1 f''(x)<0 (concave down)
and for x>1 f''(x)>0 (concave up)

since the critical point x=0<1is in a concave down region it is a relative max.