Question

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.) f(x) = 5 − 24x + 4x2, [2, 4]

Answer 1

The number
`c = 3` satisfies the conclusion of Rolle's Theorem for
`f(x) = 5-24\cdot x +4\cdot x^(2)`.

Explanation:

According to the Rolle's Theorem, for all function continuous on
`[a,b]`, there is a value
`c` (
`a \leq c\leq b`) such that:

`f'(c) = (f(b)-f(a))/(b-a)` (Eq. 1)

Where:

`f'(c)` - First derivative of the function evaluated at
`x = c`, dimensionless.

`a`,
`b` - Lower and upper bounds, dimensionless.

`f(a)`,
`f(b)` - Function evaluated at lower and upper bounds, dimensionless.

Let
`f(x) = 5-24\cdot x +4\cdot x^(2)`, then upper and lower values are, respectively:

Lower bound (
`a = 2`)

`f(2) = 5-24\cdot (2) +4\cdot (2)^(2)`

`f(2) = -27`

Upper bound (
`b = 4`)

`f(4) = 5-24\cdot (4) +4\cdot (4)^(2)`

`f(4) = -27`

From Rolle's Theorem, we find that first derivative evaluated at
`c` is:

`f'(c) = (-27-(-27))/(4-2)`

`f'(c) = 0`

Then, we find the first derivative of the function, equalize
`x` to
`c` and solve the resulting expression:

`f'(x) = -24+8\cdot x`

`-24+8\cdot c = 0`

`c = 3`

The number
`c = 3` satisfies the conclusion of Rolle's Theorem for
`f(x) = 5-24\cdot x +4\cdot x^(2)`.