a. Determine whether the Mean Value Theorem applies to the function $f(x)=-5+x^{2}$ on the interval $[-1,2]$
b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.
a. Choose the correct answer below.
A. Yes, because the function is continuous on the interval $[-1,2]$ and differentiable on the interval $(-1,2)$.
B. No, because the function is not continuous on the interval $[-1,2]$, and is not differentiable on the interval $(-1,2)$.
C. No, because the function is continuous on the interval $[-1,2]$, but is not differentiable on the interval $(-1,2)$.
D. No, because the function is differentiable on the interval $(-1,2)$, but is not continuous on the interval $[-1,2]$
b. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The point(s) is/are $\mathrm{x}=$ (Simplify your answer. Use a comma to separate answers as needed.)
B. The Mean Value Theorem does not apply in this case
Answer
Final Answer: a. The correct answer is A. Yes, because the function is continuous on the interval \([-1,2]\) and differentiable on the interval \((-1,2)\). b. The correct choice is A. The point(s) is/are \(\mathrm{x}=\boxed{\frac{1}{2}}\).
Step 1 ：The Mean Value Theorem states that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the interval (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].
Step 2 ：In this case, the function is \(f(x)=-5+x^{2}\) and the interval is \([-1,2]\). The function is a polynomial function, which is continuous and differentiable for all real numbers. Therefore, the Mean Value Theorem applies to this function on the interval \([-1,2]\).
Step 3 ：To find the point(s) that are guaranteed to exist by the Mean Value Theorem, we need to find the derivative of the function and set it equal to the average rate of change of the function over the interval \([-1,2]\). The average rate of change of the function over the interval \([-1,2]\) is \(\frac{f(2)-f(-1)}{2-(-1)}\).
Step 4 ：The derivative of the function \(f(x)=-5+x^{2}\) is \(f'(x)=2x\). Setting this equal to the average rate of change gives us \(2x=1\), which simplifies to \(x=\frac{1}{2}\).
Step 5 ：Final Answer: a. The correct answer is A. Yes, because the function is continuous on the interval \([-1,2]\) and differentiable on the interval \((-1,2)\). b. The correct choice is A. The point(s) is/are \(\mathrm{x}=\boxed{\frac{1}{2}}\).
