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Let f and g be differentiable on [0,1] such that f(0)=2, g(0)=0, g(1)=2, and f(1)=6. Then there exists some c in (0,1) for which

1. f'(c)=g'(c)
2. f'(c)=2g'(c)
3. 2f'(c)=g'(c)
4. 2f'(c)=3g'(c)

Correct answer: f'(c)=2g'(c)

Solution

The correct option is based on the application of the Mean Value Theorem, which states that if two functions are continuous and differentiable on a closed interval, then there exists at least one point in the interval where the derivatives are proportional to the changes in the function values. Here, the change in f from 2 to 6 and the change in g from 0 to 2 suggests that the derivative of f at some point is twice that of g, leading to the conclusion that f'(c) = 2g'(c).

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