Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. (Round your answer to four decimal places. Enter your answers as a comma-separated list.) f(x)=5√x,[4,9]

The possible value of c for is 6.25

The function is given as:

Calculate f(4) and f(9)

Substitute c for x in f(x)

Calculate f'(c)

So, we have:

This gives

Also, we have:

Substitute c for x

Substitute 1 for f'(c)

Multiply through by 2/5

This gives

Square both sides

Hence, the possible value of c is 6.25

Read more about mean value theorem at:

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Step-by-step explanation:

The mean value theorem for integrals for the function f(c) over a given interval [a, b] is expressed as g prime(c) = g(b) - g(a)/b-a. The idea is that there is a value c in between the interval [a, b] for the function given.

Given the function g(x) = 5√x within the interval [4,9]

g prime (c) = g(9) - g(4)/9-4

g(9) = 5√9

g(9) = 5*3 = 15

g(4) = 5√4

g(4) = 5*2 = 10

g prime c) = 15-10/9-4

g prime (c) = 5/5

g prime(c) = 1

So we are to find the number for which g prime (x) = g prime(c)

If g(x) = 5√x =

g prime (x) =

g prime (x) = 5/2√x

Since g prime (c) = 1 then;

5/2√x = 1

5 = 2√x

√x = 5/2

x = (5/2)²

x = 25/4

The value of c guaranteed by the mid value theorem is 25/4

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