let S be the set of all points in `(-pi,pi)` at which the function , f(x) =min {sinx , cos x} is not differentiable Then S is a subset of which of the following ?

Options

A. `{-(3pi)/4,-pi/4,(3pi)/4,pi/4}`
B. 1
C. 2
D. 3

Answer

To find the set \( S \) of all points in \( (-\pi, \pi) \) at which the function \( f(x) = \min(\sin x, \cos x) \) is not differentiable, we will follow these steps:

### Step 1: Identify where \( \sin x \) and \( \cos x \) intersect.
The function \( f(x) \) will change its definition at points where \( \sin x = \cos x \). We can find these points by solving the equation:
\[ \sin x = \cos x \]
This can be rewritten using the tangent function:
\[ \tan x = 1 \]
The general solutions for \( \tan x = 1 \) are:
\[ x = \frac{\pi}{4} + n\pi \quad \text{for } n \in \mathbb{Z} \]
Within the interval \( (-\pi, \pi) \), the specific solutions are:
\[ x = -\frac{3\pi}{4}, \frac{\pi}{4} \]

### Step 2: Determine the behavior of \( f(x) \) around the intersection points.
At the points \( x = -\frac{3\pi}{4} \) and \( x = \frac{\pi}{4} \), the function \( f(x) \) transitions from one function to another:
- For \( x < -\frac{3\pi}{4} \), \( \sin x < \cos x \) so \( f(x) = \sin x \).
- For \( -\frac{3\pi}{4} < x < \frac{\pi}{4} \), \( \sin x > \cos x \) so \( f(x) = \cos x \).
- For \( x > \frac{\pi}{4} \), \( \sin x < \cos x \) so \( f(x) = \sin x \).

### Step 3: Check differentiability at the intersection points.
To check differentiability at the points \( x = -\frac{3\pi}{4} \) and \( x = \frac{\pi}{4} \), we need to find the left-hand and right-hand derivatives:
- At \( x = -\frac{3\pi}{4} \):
- Left-hand derivative: \( f'(x) = \cos x \) as \( x \to -\frac{3\pi}{4}^- \)
- Right-hand derivative: \( f'(x) = -\sin x \) as \( x \to -\frac{3\pi}{4}^+ \)

Since the left-hand and right-hand derivatives are not equal, \( f(x) \) is not differentiable at \( x = -\frac{3\pi}{4} \).

- At \( x = \frac{\pi}{4} \):
- Left-hand derivative: \( f'(x) = -\sin x \) as \( x \to \frac{\pi}{4}^- \)
- Right-hand derivative: \( f'(x) = \cos x \) as \( x \to \frac{\pi}{4}^+ \)

Again, since the left-hand and right-hand derivatives are not equal, \( f(x) \) is not differentiable at \( x = \frac{\pi}{4} \).

### Conclusion:
The set \( S \) of points in \( (-\pi, \pi) \) where \( f(x) \) is not differentiable is:
\[ S = \left\{ -\frac{3\pi}{4}, \frac{\pi}{4} \right\} \]