Given the function #f(x) = (x)/(x+2)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,4] and find the c?
Is
The Mean Value Theorem has two hypotheses:
This function on this interval satisfies the hypotheses of the Mean Value Theorem.
Therefore, we know without any further work that
We need to solve
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To determine if the function satisfies the hypotheses of the Mean Value Theorem on the interval [1, 4], you need to check if the function is continuous on the closed interval [1, 4] and differentiable on the open interval (1, 4).

Check continuity: Evaluate the function at the endpoints of the interval, f(1) and f(4), to ensure it's continuous on the closed interval.
 f(1) = 1/(1+2) = 1/3
 f(4) = 4/(4+2) = 4/6 = 2/3

Check differentiability: Calculate the derivative of the function and check if it exists on the open interval (1, 4).
 f'(x) = d/dx (x/(x+2)) = [(x+2)(1)  (x)(1)] / (x+2)^2 = 2/(x+2)^2
The function f(x) = x/(x+2) is continuous and differentiable on the interval (1, 4).
Now, to find the c that satisfies the Mean Value Theorem, apply the formula:
 f'(c) = [f(4)  f(1)] / (4  1)
Substitute the values:
 2/(c+2)^2 = (2/3  1/3) / 3
 2/(c+2)^2 = 1/9
Solve for c:
 (c+2)^2 = 18
 c + 2 = ±√18
 c = 2 ± √18
Since the interval is [1, 4], the value of c must be in the range (1, 4). Therefore, we discard the negative root.
 c = 2 + √18 ≈ 1.764
So, the value of c that satisfies the Mean Value Theorem on the interval [1, 4] is approximately 1.764.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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