Question
Use Cauchy's mean value theorem to prove that:
Answer :
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Consider the functions (f(x) = \cos x) and (g(x) = \sin x), which are continuous on the closed interval ([\alpha, \beta]) and differentiable on the open interval ((\alpha, \beta)), where (0 < \alpha < \theta < \beta < \frac{\pi}{2}). According to Cauchy's Mean Value Theorem, there exists a point (\theta \in (\alpha, \beta)) such that: [ \frac{f'(\theta)}{g'(\theta)} = \frac{f(\beta) - f(\alpha)}{g(\beta) - g(\alpha)}. ] Now, compute the derivatives: [ f'(x)
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