Question

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)

1. 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.