Proving Derivability: A Comprehensive Guide

When tackling mathematical problems involving the comparison of functions and their derivatives, it is crucial to utilize the appropriate theorems and lemma. In this article, we will explore the proof of a specific derivability question and demonstrate how to apply Rolle's Lemma. This guide is intended for students and professionals dealing with mathematical analysis and optimization, ensuring a clear and structured approach to solving similar problems.

Introduction to the Problem

Consider the functions (f(x)) and (g(x)) which are continuous on the interval ([-1, 1]) and differentiable on ((-1, 1)). Additionally, given that (f(1) g(1)) and (f(-1) g(-1)), we need to prove that there exists a point (c) in the interval ((-1, 1)) such that the derivatives of (f(x)) and (g(x)) are equal at this point, i.e., (f'(c) g'(c)).

Utilizing Rolle's Lemma

Rolle's Lemma is a fundamental theorem in calculus that states: if a function (h(x)) is continuous on the closed interval ([a, b]) and differentiable on the open interval ((a, b)), and if (h(a) h(b)), then there is at least one point (c) in ((a, b)) such that (h'(c) 0). This lemma is particularly useful in the context of proving the existence of a point (c) where the derivatives of two functions are equal.

To apply Rolle's Lemma, let us define a new function:

[h(x) f(x) - g(x)]

Note that (h(x)) is continuous on ([-1, 1]) and differentiable on ((-1, 1)) because (f(x)) and (g(x)) share these properties. Additionally, given the conditions (f(1) g(1)) and (f(-1) g(-1)), we have:

[h(1) f(1) - g(1) 0]

[h(-1) f(-1) - g(-1) 0]

Since (h(x)) is continuous on ([-1, 1]) and differentiable on ((-1, 1)), and (h(1) h(-1) 0), we can apply Rolle's Lemma directly. According to Rolle's Lemma, there exists at least one point (c) in the open interval ((-1, 1)) such that:

[h'(c) 0]

Since (h(x) f(x) - g(x)), we have:

[h'(x) f'(x) - g'(x)]

Therefore, at the point (c), we get:

[h'(c) f'(c) - g'(c) 0]

Thus, we have proven that there exists a point (c) in ((-1, 1)) such that:

[f'(c) g'(c)]

Proving Rolle's Lemma from First Principles

For the sake of completeness, let's briefly outline the proof of Rolle's Lemma from first principles. Consider the function (h(x)) that is continuous on ([a, b]) and differentiable on ((a, b)) with (h(a) h(b) 0).

First, if (h(x) equiv 0) for all (x) in ([a, b]), then trivially, (h'(x) 0) for all (x) in ((a, b)). Otherwise, there exists at least one point (c) in ((a, b)) where (h(x)) attains a local maximum or minimum. By Fermat's Theorem, at this point (c), we have:

[h'(c) 0]

This completes the proof of Rolle's Lemma.

Conclusion

In conclusion, by defining (h(x) f(x) - g(x)) and applying Rolle's Lemma, we have proven that there exists a point (c) in the interval ((-1, 1)) where the derivatives of (f(x)) and (g(x)) are equal. This approach not only demonstrates the utility of Rolle's Lemma but also provides a solid foundation for understanding and solving similar problems in mathematical analysis.

For further reading, Rolle's theorem is a well-documented topic. You can explore more about it on Wikipedia. Understanding this theorem will equip you with a valuable tool for proving the existence of critical points and equating derivatives in a wide range of scenarios.