Question

1. Verify that the function

satisfies the three
hypotheses of Rolle's
Theorem on the given
interval. Then find all
numbers c that satisfies the
conclusion of Rolle's
The orem.
y=-x^3+ 4x^2 – 3 ; [0, 4]

Answer 1

0 and 8/3

Explanation:

y is continuous on [0,4]

It also also differentiable on (0,4)

Plugging in 0 and 4, we get -3 both times so y(0) = y(4).

Therefore there is at least one value c between 0 and 4 inclusive where y'(c) = 0

y' = -3x^2 + 8x

Set that to 0:

-3x^2 + 8x = 0

x(-3x+8) = 0

x = 0 or 8/3

And both values work.