Question

What value of c satisfies the conclusion of the mean value theorem for f(x) = (x - 1)^2 over the interval [0, 5]?

2.5
4
8.5
10

Answer 1

The value of c that satisfies the conclusion of the mean value theorem for f(x) = (x - 1)² over the interval [0, 5] is 5.

Step-by-step explanation:

The mean value of a function over an interval is given by the formula:

Mean value = (f(b) - f(a))/(b - a)

Here, the function is f(x) = (x - 1)², the interval is [0, 5], and we need to find the value of c that satisfies the conclusion of the mean value theorem. Using the mean value theorem, we can set up the equation:

((c - 1)² - 0)/(c - 0) = (5 - 1)²/(5 - 0)

Simplifying this equation, we get:

(c - 1)² = 16

Taking the square root of both sides, we get:

c - 1 = ±4

Therefore, the possible values of c are 1 + 4 = 5 and 1 - 4 = -3. However, since c must lie within the interval [0, 5], the only valid value is c = 5.