Determine the intervals on which the function f(x) = (x^4) − (8x^3) − (72x^2) + 24x is concave up.

Question

Determine the intervals on which the function f(x) = (x^4) − (8x^3) − (72x^2) + 24x is concave up.

Answer 1

To determine concave up we need to find where the second derivative is positive, so let's find the 2nd derivative:

f(x) = (x^4) − (8x^3) − (72x^2) + 24x

f'(x) = 4x³ - 24x² - 144x + 24

f''(x) = 12x² - 48x - 144

Now let's find where the equation is >0:

12(x²-4x-12)>0, (x-6)(x+2)>0,

Using the result we find x≤-2 or x≥6, (-infinity,-2)U(6,infinity)