Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. f(x) = x, [0, 9]

Solution:

According to the Mean Value Theorem,

Suppose y = f(x) is continuous on a closed interval [a, b] and differentiable on the interval’s interior (a, b).

Then there is at least one point c in (a, b) at which

(f(b) - f(a))/(b - a) = f’(c)

Now f(x) = x is continuous in the interval (0, 9)

f’(c) = (f(b) - f(a))/(b - a)

= (f(9) - (f(0))/(9 - 0)

= (9 - 0)/(9 - 0)

= 1

Therefore f’(c) = 1

Since f(x) = x and f’(x) = 1 and hence

f’(c) = 1

Therefore c = 1

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. f(x) = x, [0, 9]

Summary:

The number c that satisfies the conclusion of the Mean Value Theorem on the given interval. f(x) = x, [0, 9] is 1.