Given the function #f(x)=x/(x+9)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,4] and find the c?

The Mean Value Theorem has two Hypotheses. A hypothesis is satisfied if it is true.

To determine if the function f(x) satisfies the hypotheses of the Mean Value Theorem on the interval [1,4], we need to check if two conditions are met:

1. The function f(x) must be continuous on the closed interval [1,4].
2. The function f(x) must be differentiable on the open interval (1,4).

Firstly, f(x) = x/(x+9) is continuous for all real numbers except where the denominator is zero. Since x+9 ≠ 0 for all real numbers, f(x) is continuous everywhere. Therefore, it is continuous on the closed interval [1,4].

Next, to check differentiability, we need to ensure that the derivative of f(x) exists for all x in the open interval (1,4). The derivative of f(x) can be found using the quotient rule:

f'(x) = (1*(x+9) - x*(1))/(x+9)^2 = 9/(x+9)^2

This derivative is defined for all x ≠ -9, which is true for all x in the interval (1,4).

Since both conditions are met, f(x) satisfies the hypotheses of the Mean Value Theorem on the interval [1,4].

To find the c value guaranteed by the Mean Value Theorem, we use the formula:

f'(c) = (f(4) - f(1))/(4 - 1)

Plugging in the values:

f'(c) = (4/(4+9) - 1/(1+9))/(4 - 1) = (4/13 - 1/10)/3

Solving this equation will give us the value of c.