Question

what is the relative maximum and minimum of the function? f(x)=x^3+6x^2 -36x

Answer 1

The max/mins occur when the derivative is 0.\[f'(x)=3x^2+12x-36=0\]Divide by 3 to simplify.\[x^2+4x-12=0\]Factor.\[(x+6)(x-2)=0\]So max/mins may occur when x=-6 or x=2.\[f(-6)=-216+216+216=216\]\[f(2)=8+24-72=-40\]If you graph it or check for where f(x) is increasing/decreasing you can tell that f(-6)=216 is a relatie maximum and f(2)=-40 is a relative minimum.