If f(x) and g(x) are differentiable functions for 0 ≤x ≤1 such that f(0)=2, g(0)=0, f(1)=6, g(1)

step 1 In the question we are given that value of f0 equal to 2 value of z 0 equal to 0 value of f1 equ

If f(x) and g(x) are differentiable functions for 0 ≤x ≤1...

If $f(x)$ and $g(x)$ are differentiable functions for $0 \leq x \leq 1$ such that $f(0)=2, g(0)=0, f(1)=6, g(1)=2$, then in the interval $(0,1)$, (A) $f^{\prime}(x)=0$ for all $x$ (B) $f^{\prime}(x)=2 g^{\prime}(x)$ for at least one $x$ (C) $f^{\prime}(x)=2 g^{\prime}(x)$ for at most one $x$ (D) None of these

If $f(x)$ and $g(x)$ are differentiable functions for $0 \leq x \leq 1$ such that $f(0)=2, g(0)=0, f(1)=6, g(1)=2$, then in the interval $(0,1)$,
(A) $f^{\prime}(x)=0$ for all $x$
(B) $f^{\prime}(x)=2 g^{\prime}(x)$ for at least one $x$
(C) $f^{\prime}(x)=2 g^{\prime}(x)$ for at most one $x$
(D) None of these

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Key Concepts

Mean Value Theorem

The Mean Value Theorem is a fundamental result in calculus which guarantees that for any function that is continuous on a closed interval and differentiable on the open interval, there exists at least one point in the open interval where the instantaneous rate of change (derivative) equals the average rate of change over the interval. This theorem provides a bridge between the behavior of a function over an interval and its derivative at some interior point.

Cauchy's Mean Value Theorem

Cauchy's Mean Value Theorem extends the Mean Value Theorem to the case of two functions that are continuous on a closed interval and differentiable on the open interval. It asserts that if one function does not have a zero derivative when the other has a zero difference over the interval, then there exists a point in the interval where the ratio of the derivatives equals the ratio of the differences of the function values at the endpoints. This is particularly useful when comparing the rates of change of two functions over an interval.

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