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Answer :
The Mean Value Theorem states that if a function f is continuous on the interval [a,b] and differentiable on (a,b), then there exists at least one c in the interval (a,b) such that
f'(c) = (f(b) - f(a)) / (b - a)
For the function f(x) = x/x+2, we can set
f(a) = a/a+2 and f(b) = b/b+2
Therefore, the equation we need to solve is:
(b/b+2 - a/a+2) / (b - a) = c/(c+2)
After rearranging, we get:
(b-a)c + 2(a-b) = 0
Solving this, we find that c = 2(b-a) / (a-b)
Therefore, the values of c that satisfy the Mean Value Theorem for the given function f(x) = x/x+2 are c = 2(b-a) / (a-b).
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