verify that the function satisfies the hypotheses of the mean values theorem on the given interval. then find all numbers c in the given interval that satisfy the conclusion of the mean values theorem

f(x)= 2x/x-4 [0,3]

To verify if the function satisfies the hypotheses of the Mean Value Theorem (MVT), we need to check if the following conditions are met:

1. Continuity: The function must be continuous on the closed interval [0, 3].
2. Differentiability: The function must be differentiable on the open interval (0, 3).

Let's check each condition one by one:

1. Continuity:
To determine if the function is continuous on the closed interval [0, 3], we need to check if it is continuous at every point within this interval, including the endpoints.

For f(x) = 2x/(x - 4), we need to check the continuity at x = 0 and x = 3.

At x = 0:
f(0) = 2(0)/(0 - 4) = 0/(-4) = 0.

At x = 3:
f(3) = 2(3)/(3 - 4) = 6/(-1) = -6.

Since the function is a rational function and the denominator is not zero, f(x) is continuous at both x = 0 and x = 3.

2. Differentiability:
To determine if the function is differentiable on the open interval (0, 3), we need to check if its derivative exists and is continuous within this interval.

To find the derivative of f(x), we use the quotient rule:

f'(x) = [2(x - 4) - 2x]/(x - 4)^2
= (-2(4))/((x - 4)^2)
= -8/(x - 4)^2

The derivative is defined for all values of x except x = 4. Since the interval (0, 3) does not include x = 4, the derivative of f(x) is continuous within this interval.

Therefore, both continuity and differentiability conditions are satisfied for f(x) = 2x/(x - 4) on the interval [0, 3]. Hence, the function satisfies the hypotheses of the Mean Value Theorem.

To find all numbers c in the given interval [0, 3] that satisfy the conclusion of the Mean Value Theorem, we can use the formula:

f'(c) = (f(3) - f(0))/(3 - 0)

Plugging in the values we calculated earlier:
f'(c) = (-6 - 0)/(3 - 0)
= -6/3
= -2

Now, we can find c by solving for f'(c) = -2:

-8/(c - 4)^2 = -2

To solve this equation, we can cross-multiply:

-8 = -2(c - 4)^2
Divide both sides by -2:
4 = (c - 4)^2

Taking the square root of both sides:
2 = c - 4

Solving for c:
c = 2 + 4
c = 6

Therefore, the number c in the interval [0, 3] that satisfies the conclusion of the Mean Value Theorem is c = 6.