Using the function y= 1/3x^3
a) Verify that the function satisfies the hypotheses of the Mean Value Theorem on the interval [-3,3].
b) Then find all numbers c that satisfy the conclusion of the Mean Value Theorem.
To verify whether a function satisfies the hypotheses of the Mean Value Theorem (MVT), we need to check two conditions:
1. Continuity: The function must be continuous on the closed interval [-3, 3].
2. Differentiability: The function must be differentiable on the open interval (-3, 3).
Let's first check the continuity condition:
The function y = (1/3)x^3 is a polynomial function, and all polynomial functions are continuous over their entire domain. Hence, the function is continuous on the closed interval [-3, 3].
Now, let's check the differentiability condition:
To determine if a function is differentiable, we need to calculate its derivative. The derivative of the function y = (1/3)x^3 can be found by differentiating term by term:
dy/dx = d/dx ((1/3)x^3)
= (1/3) * d/dx (x^3)
= (1/3) * 3x^2
= x^2
The derivative of y = (1/3)x^3 is given by dy/dx = x^2. The derivative exists for all real numbers, and hence, it exists on the open interval (-3, 3).
Since the function y = (1/3)x^3 satisfies both conditions (continuity and differentiability), we can conclude that it satisfies the hypotheses of the Mean Value Theorem on the interval [-3, 3].
Now, let's move to the second part of the question and find the values of c that satisfy the conclusion of the Mean Value Theorem.
According to the Mean Value Theorem, for a function that satisfies its hypotheses on a closed interval [a, b], there exists at least one value c in the open interval (a, b) such that the derivative at c is equal to the average rate of change of the function over [a, b].
The average rate of change of the function y = (1/3)x^3 over the interval [-3, 3] is given by:
Average rate of change = (f(b) - f(a))/(b - a)
For b = 3 and a = -3, we have:
f(3) = (1/3)(3)^3 = 9
f(-3) = (1/3)(-3)^3 = -9
Plugging these values into the average rate of change formula, we get:
Average rate of change = (9 - (-9))/(3 - (-3))
= 18/6
= 3
Now, we need to find the values of c for which the derivative of the function, dy/dx = x^2, is equal to 3.
To solve this equation, we set x^2 = 3 and solve for x.
x^2 = 3
Taking the square root on both sides:
x = ±√3
So, the values of c that satisfy the conclusion of the Mean Value Theorem are c = √3 and c = -√3.
Thus, there exist at least two values, c = √3 and c = -√3, in the interval (-3, 3) for which the instantaneous rate of change (derivative) at those points equals the average rate of change of the function over the interval [-3, 3].