Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval.

f(x) = 4√x

[4, 9]

f(4) = 8

f(9) = 12
slope of secant = 4/5

f'(x) = 2/√x

so, find c such that

2/√c = 4/5

To find the value(s) of c guaranteed by the Mean Value Theorem for Integrals, we first need to verify if the function f(x) satisfies the three requirements of the Mean Value Theorem (MVT) for Integrals:

1. Continuity: The function f(x) = 4√x is continuous on the closed interval [4, 9] since it is a square root function and there are no breaks or discontinuities within this interval.

2. Integrability: The function f(x) = 4√x is integrable on the interval [4, 9] since it is a well-defined function without any vertical asymptotes or undefined points within this interval.

3. Average value: We also need to calculate the average value of the function over the interval [4, 9], denoted by "f(avg)". The average value of a continuous function f(x) over the interval [a, b] is given by the formula:

f(avg) = 1/(b - a) ∫[a, b] f(x) dx

In this case, a = 4, b = 9, and f(x) = 4√x. Plugging these values into the formula, we have:

f(avg) = 1/(9 - 4) ∫[4, 9] 4√x dx
= 1/5 ∫[4, 9] 4√x dx
= 4/5 ∫[4, 9] √x dx

To integrate √x, let's use the power rule of integration:

∫ x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1.

Applying this rule with n = 1/2, we have:

∫ √x dx = (2/3)√x^3/2 + C
= (2/3)√x^3/2 + C

Now we can find the integral over the interval [4, 9]:

∫[4, 9] √x dx = [(2/3)√x^3/2] evaluated from 4 to 9
= [(2/3)√(9^3/2)] - [(2/3)√(4^3/2)]
= (2/3)√243 - (2/3)√64
= (2/3)(9√3) - (2/3)(8)
= 6√3 - (16/3)

Therefore, the average value of f(x) = 4√x over the interval [4, 9], f(avg), is:

f(avg) = 4/5 * (6√3 - (16/3))
= (24√3 - 64/5)/5
= (24√3 - 64/5)/5
≈ -0.313307

Now, according to the Mean Value Theorem for Integrals, there exists at least one value c in the interval [4, 9] such that the definite integral of the function f(x) over this interval is equal to f(avg):

∫[4, 9] 4√x dx = f(avg) * (9 - 4)

To find the value(s) of c, we set up the equation and solve for c:

∫[4, 9] 4√x dx = -0.313307 * 5
4 * [(2/3)√x^3/2] evaluated from 4 to c = -1.566535

Substituting the limits of integration:

4 * [(2/3)√c^3/2] - [4 * (2/3)√(4^3/2)] = -1.566535

Let's simplify this equation:

(8/3)√c^3/2 - (8/3)√(4^3/2) = -1.566535

Now, we can solve for c.