Critical Points on a Curve?

📌 CONCEPT: A function f(x) is differentiable at a point x = a if the function has a unique tangent line at that point, i.e., the limit of the difference quotient exists as x approaches a.

📐 RULE / FORMULA: The function f(x) = |x^2 - 4x| + 1 is differentiable at all points except at x = 2 and x = 0, where the function has a corner or a cusp.

💡 WORKED EXAMPLE: To show that f(x) is differentiable at x = 2, we need to prove that the limit of the difference quotient exists as x approaches 2. Let's compute the left-hand and right-hand derivatives at x = 2. We have f(x) = |x^2 - 4x| + 1, and the left-hand derivative at x = 2 is lim(h → 0) [f(2 + h) - f(2)]/h, which exists. Similarly, the right-hand derivative also exists. Since the left-hand and right-hand derivatives are equal, f(x) is differentiable at x = 2.

⚠️ COMMON MISTAKE: Students often assume that a function is differentiable at all points where the function is continuous, but this is not always true. A function can be continuous at a point without being differentiable, as seen in this example where f(x) has a corner at x = 2 and a cusp at x = 0.