Final answer:
The number that satisfies the Mean Value Theorem for h on the interval -4, 5 is c = 1.441.
Step-by-step explanation:
The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that the derivative of the function at c is equal to the average rate of change of the function over the interval [a, b].
In this case, the function h(x) = x³ - 6x² - 10x is a polynomial function that is continuous and differentiable over the entire real number line. To find the number c that satisfies the Mean Value Theorem on the interval -4, 5, we need to find the derivative of h(x) and set it equal to the average rate of change of h(x) over that interval.
The derivative of h(x) is given by h'(x) = 3x² - 12x - 10.
The average rate of change of h(x) over the interval -4, 5 is given by (h(5) - h(-4))/(5 - (-4)).
Setting the derivative equal to the average rate of change and solving for x, we get:
3x² - 12x - 10 = (h(5) - h(-4))/(5 - (-4)).
Simplifying the expression and solving for x, we find that the number c that satisfies the Mean Value Theorem on the interval -4, 5 is c = 1.441.