Limits, Continuity & Differentiability
Continuity And Differentiability
nta_abhyas_2025
Grade 12
Question:
In $(0, 2\pi)$, the total number of points where $f(x) = \max(\sin x, \cos x, 1 - \cos x)$ is not differentiable, are equal to
Step-by-Step Solution
Key Concept: A function defined as the maximum of several smooth functions is not differentiable at points where the active piece changes.
The graph of $f(x) = \max(\sin x, \cos x, 1 - \cos x)$ shows that $f(x)$ is not differentiable at points where the maximum function switches from one expression to another. From the graph, these points of non-differentiability occur at $x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}$ (and potentially other points depending on the domain shown). The solution indicates that $f(x)$ is not differentiable at $3$ points within the considered interval.
Correct Answer: 3