If the functions f and g are differentiable functions on...

If the functions $f$ and $g$ are differentiable functions on $[0,1]$ satisfying $f(0)=2=g(1), g(0)=0$ and $f(1)=6$,
then for some $c \in] 0,1[$
(A) $2 f^{\prime}(C)=g^{\prime}(c)$
(B) $2 f^{\prime}(C)=3 g^{\prime}(c)$
(C) $f^{\prime}(C)=g^{\prime}(c)$
(D) $f^{\prime}(C)=2 g^{\prime}(c)$

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Key Concepts

Differentiability

Differentiability is a key concept ensuring that a function has a derivative at every point in an interval. It guarantees the smooth behaviour of functions and is a prerequisite for applying powerful theorems in calculus that relate the function's overall change to its instantaneous rate of change.

Mean Value Theorem

The Mean Value Theorem (MVT) states that for any function that is continuous on a closed interval and differentiable on the open interval, there exists at least one point at which the derivative (instantaneous rate of change) equals the average rate of change over that interval. This theorem is fundamental in linking local behaviour (derivatives) with global behaviour (net change) of functions.

Cauchy's Mean Value Theorem

Cauchy's Mean Value Theorem is a generalization of the Mean Value Theorem that applies to two functions simultaneously. It establishes that if two functions are continuous on a closed interval and differentiable on the open interval (with a nonzero derivative for one of them at some point), there exists a point in the interval where the ratio of their derivatives equals the ratio of their total changes over the interval. This theorem is especially useful for comparing the rates of change of two functions.

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