Question

I'm having problems solving this and showing the steps:

Find the absolute maximum and absolute minimum, if either exists, for

y = f(x)= x^3 - 12x + 12 -3 less equal to x greater than equal to 5

Answer 1

I'll rewrite the equation.
y = f(x) = x^3 -12x + 12
-3 <= x >= 5

Answer 2

dy/dx = 3x^2 -12
function is horizontal when dy/dx = 0
that is when
x^2 = 4
x = +/- 2
when x = -2
f(-2) = -8 + 24 + 12 = 28
and
d^2y/dx^2 = 6 x = -12 so a maximum
when x = 2
f(2) = -4
and
d^2y/dx^2 = 6x = +12 so a minimum
now check the end points
when x = -3
f(-3) = -27+36+12 = 21
when x=+5
f(5) = 125 - 60 +12 = 77
so the maximum is 77 at x = 5
and the minimum is -4 at x = 2

Answer 3

Thank you so much. This is a great help. I can see where I started to go wrong.