determine all critical points for the function

Question

determine all critical points for the function f(x)= x(4-x)^3

Answer 1

To determine critical points, you first have to find the derivative and then set that derivative equal to zero.

f(x)= x(4-x)³

We can use the product rule for this because x is multiplied by (4-x)³ , so

f'(x) = (4-x)³ * x' + x [(4-x)³]'

Keep in mind the derivative for (4-x)³ uses the chain rule, so 3u² * u'

= (4-x)³ * 1 + x (-1 * 3(4-x)²)

= (4-x)³ - 3x (4-x)²

We can factor out a (4-x)²

(4-x)² [(4-x) - 3x] = 0

So our two possible equations are (4-x)² = 0 and 4 - 4x = 0

This gives us two critical points, x = 4 or x = 1.