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]

The number satisfies the conclusion of Rolle's Theorem for .

Step-by-step explanation:

According to the Rolle's Theorem, for all function continuous on , there is a value () such that:

(Eq. 1)

Where:

- First derivative of the function evaluated at , dimensionless.

, - Lower and upper bounds, dimensionless.

, - Function evaluated at lower and upper bounds, dimensionless.

Let , then upper and lower values are, respectively:

Lower bound ()

Upper bound ()

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

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

The number satisfies the conclusion of Rolle's Theorem for .

the answer is c. 23/3