Question

question 7 (3 points)

for \\(f(x) = x^2\\) on the interval \\(1, 3\\), which value of \\(c\\) satisfies the conclusion of the mean value theorem?

\\(c = 1\\)
\\(c = 2\\)
\\(c = 3\\)
no such \\(c\\) exists

Explanation:

Calculate the average rate of change

Using the Mean Value Theorem knowledge point

$$ \frac{f(3) - f(1)}{3 - 1} = \frac{3^2 - 1^2}{3 - 1} = \frac{9 - 1}{2} = 4 $$

Find the derivative of the function

Using the Derivative of Polynomials knowledge point

$$ f'(x) = 2x $$

Solve for the value of c

Using the Mean Value Theorem knowledge point

$$
LATEXBLOCK0
$$

Since \(c = 2\) lies in the open interval \((1, 3)\), it satisfies the theorem.

Answer:

• \(c = 1\)
• \(c = 2\) (Correct answer)
• \(c = 3\)
• No such \(c\) exists