# 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?

Please see below.

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

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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:

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

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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