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)

Answers

The value of c by the Mean Value Theorem for Integrals for the function over the given integral is 6.25.

In the question, we are asked to find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function f(x)=5√x,[4,9] over the given interval.

The given function is f(x)=5√x,[4,9].

First we calculate the function value at the endpoints of the interval.

f(4) = 5√4 = 5*2 = 10.

f(9) = 5√9 = 5*3 = 15.

We substitute x = c in the given function, to get:

f(c) = 5√c.

Now, we calculate f'(c) using the formula:

f'(c) = (f(b) – f(a))/(b – a),

or, f'(c) = (f(9) – f(4))/(9 – 4),

or, f'(c) = (15 – 10)(9 – 4),

or, f'(c) = 5/5 = 1.

Differentiating the original function, gives us f'(x) = 5/(2√x).

Substituting x = c in this gives us f'(c) = 5/(2√c).

Equating this to f'(c) = 1 gives us:

5/(2√c) = 1,

or, √c = 5/2,

or, c = 25/4,

or, c = 6.25.

Thus, the value of c by the Mean Value Theorem for Integrals for the function over the given integral is 6.25.

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“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.)

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