Differentiability at Critical Points?

📌 CONCEPT: A function is said to be differentiable at a point if it has a tangent line at that point, and the function's derivative exists at that point.

📐 RULE / FORMULA: Rolle's theorem states that if a function f(x) is continuous on the interval [a, b] and differentiable on the interval (a, b), then there exists a point c in (a, b) such that f'(c) = 0.

💡 WORKED EXAMPLE: Consider the function f(x) = |x^2 - 4|. It is known that f(x) is differentiable everywhere except at its critical points. Suppose there is a critical point c where f(x) is not differentiable. By Rolle's theorem, there exists a point d in (c - h, c + h) such that f'(d) = 0, where h is a small positive value. However, this contradicts the fact that f(x) is not differentiable at c. Hence, f(x) must be differentiable at all points except its critical points.

⚠️ COMMON MISTAKE: Students often mistakenly apply Rolle's theorem to a function that is not continuous on the interval [a, b], leading to incorrect conclusions about differentiability.