f(x) and g(x) are two differentiable functions, for x in [a, b] such that f(a)=3 ; g(a)=-3 ; f(b

step 1 we have problem number 171 we have been given two functions f x and g x and there are various re

F(x) and g(x) are two differentiable functions, for x in [a,...

$\mathrm{f}(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$ are two differentiable functions, for $\mathrm{x}$ in $[\mathrm{a}, \mathrm{b}]$ such that $\mathrm{f}(\mathrm{a})=3 ; \mathrm{g}(\mathrm{a})=-3 ; \mathrm{f}(\mathrm{b})=30 ; \mathrm{g}(\mathrm{b})=6$. Prove that there exists a point c satisfying $a<c<b$ at which the ratio of the derivatives of $f$ and $g$ is $3: 1$.

$\mathrm{f}(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$ are two differentiable functions, for $\mathrm{x}$ in $[\mathrm{a}, \mathrm{b}]$ such that $\mathrm{f}(\mathrm{a})=3 ; \mathrm{g}(\mathrm{a})=-3 ; \mathrm{f}(\mathrm{b})=30 ; \mathrm{g}(\mathrm{b})=6$. Prove that
there exists a point c satisfying $a<c<b$ at which the ratio of the derivatives of $f$ and $g$ is $3: 1$.

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Key Concepts

Mean Value Theorem

The Mean Value Theorem states that for any function that is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one point c in (a, b) at which the derivative of the function equals the average rate of change over [a, b]. This theorem is a cornerstone in differential calculus and connects local properties of a function (its derivative) to its overall behavior on an interval.

Cauchy's Mean Value Theorem

Cauchy's Mean Value Theorem generalizes the Mean Value Theorem to two functions. It states that if two functions are continuous on [a, b] and differentiable on (a, b), with the derivative of the second function never zero, then there exists a point c in (a, b) such that the ratio of their derivatives equals the ratio of the differences in their values at the endpoints. This theorem provides a method to relate the rates of change of two functions and is key to proving that the derivatives of f and g have a specific ratio at some point in the interval.

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