Limits, Continuity & Differentiability
Non-differentiability points
Grade 12
Question:
<p>The total number of points of non-differentiability of \(f(x) = \max\left\{\sin^2 x, \cos^2 x, \frac{3}{4}\right\}\) in \([0, 10\pi]\), is</p>
<p>(a) 40</p>
<p>(b) 30</p>
<p>(c) 20</p>
<p>(d) 10</p>
Step-by-Step Solution
Key Concept: The function f(x) is a maximum of three functions. Points of non-differentiability occur where the maximum switches from one function to another. These occur at regular intervals determined by the periodicity of sin²x and cos²x.
<p><strong>Solution:</strong> Here, $f(x) = \max\left\{\sin^2 x, \cos^2 x, \frac{3}{4}\right\}$</p><p>Since $\sin^2 x$ and $\cos^2 x$ are periodic with period $\pi$ and in $[0, \pi]$, there are four points of non-differentiability of $f(x)$.</p><p>In $[0, 10\pi]$, there are 40 points of non-differentiability.</p><p>∴ Answer is (a).</p>
Correct Answer: a