Limits, Continuity & Differentiability
Methods of Differentiation
Grade 12
Question:
<p>Let $S$ be the set of all points in $(-\pi,\pi)$ at which the function $f(x)=\min(\sin x,\cos x)$ is not differentiable. Then $S$ is a subset of which of the following?</p>
<p>$\left\{-\dfrac{3\pi}{4},-\dfrac{\pi}{4},\dfrac{3\pi}{4},\dfrac{\pi}{4}\right\}$</p>
<p>$\left\{-\dfrac{\pi}{4},0,\dfrac{\pi}{4}\right\}$</p>
<p>$\left\{-\dfrac{3\pi}{4},0,\dfrac{3\pi}{4}\right\}$</p>
<p>$\left\{-\dfrac{3\pi}{4},-\dfrac{\pi}{4},\dfrac{\pi}{4},\dfrac{3\pi}{4}\right\}$</p>
Step-by-Step Solution
Key Concept: General
<b>Non-Differentiability of $\min$ Functions</b><br>
$f(x)=\min(\sin x,\cos x)$: non-differentiable where $\sin x=\cos x$ (crossover points) — i.e., $x=\pi/4+n\pi$.<br>
In $(-\pi,\pi)$: $x=\pi/4$ and $x=-3\pi/4$. Also check $x=-\pi/4$ and $x=3\pi/4$ where $\sin x=\cos x$ again.<br>
$\sin x=\cos x\Rightarrow\tan x=1\Rightarrow x=\pi/4,-3\pi/4$ in $(-\pi,\pi)$.<br>
Wait: $\tan x=1$ has solutions $x=\pi/4$ and $x=\pi/4-\pi=-3\pi/4$ in $(-\pi,\pi)$.<br>
So $S=\{-3\pi/4,\pi/4\}\subset\{-3\pi/4,-\pi/4,\pi/4,3\pi/4\}=$ option (4). <b>Answer: 4</b><br>
<b>Key concept:</b> $\min(f,g)$ is non-differentiable where $f=g$ AND the functions cross (not just touch).<br>
<b>Trap:</b> Including all four points — only two actual crossing points exist in $(-\pi,\pi)$.
Correct Answer: 4