Question

Let f(x) = x³+6x²+6x and let c be the number that satisfies the Mean Value Theorem for f on the interval [-6,0].Find the Mean Value Theorem.

Answer 1

c = 0 or c = -4

Explanation:

According to the Mean Value Theorem

`f^(')(c) = (f(0) - f(-6))/(0 - (-6)) = (0 - (-6)^3 - 6*(-6)^2 - 6*(-6))/(6) = 36 - 36 + 6 = 6`

By taking the first derivative of f

`f^(')(x) = 3x^2 + 12x + 6`

Since
`f^(')(c) = 6`

We can solve for c

`3c^2 + 12c + 6 = 6`

`3c^2 + 12c = 0`

`c(c + 4) = 0`

c = 0 or c = -4