Let `f(x)=m ax{2 sin x,1-cosx},x epsilon (0,pi)`. Then set of points of non differentiability is

Options

A. `phi`
B. `{(pi)/2}`
C. `{pi-"cos"^(-1)3/5}`
D. `{"cos"^(-1)3/5}`

Answer

To find the set of points of non-differentiability for the function \( f(x) = \max(2 \sin x, 1 - \cos x) \) for \( x \in (0, \pi) \), we will follow these steps:

### Step 1: Identify the two functions
The function \( f(x) \) is defined as the maximum of two functions:
1. \( g_1(x) = 2 \sin x \)
2. \( g_2(x) = 1 - \cos x \)

### Step 2: Find the points of intersection
To determine where \( f(x) \) may not be differentiable, we need to find the points where \( g_1(x) = g_2(x) \):
\[ 2 \sin x = 1 - \cos x \]

### Step 3: Rearranging the equation
Rearranging the equation gives:
\[ 2 \sin x + \cos x - 1 = 0 \]

### Step 4: Solve the equation
To solve \( 2 \sin x + \cos x - 1 = 0 \), we can express \( \sin x \) in terms of \( \cos x \):
1. Isolate \( \cos x \):
\[ \cos x = 1 - 2 \sin x \]
2. Substitute \( \cos^2 x + \sin^2 x = 1 \):
\[ (1 - 2 \sin x)^2 + \sin^2 x = 1 \]
Expanding this:
\[ 1 - 4 \sin x + 4 \sin^2 x + \sin^2 x = 1 \]
Simplifying:
\[ 5 \sin^2 x - 4 \sin x = 0 \]
Factoring out \( \sin x \):
\[ \sin x (5 \sin x - 4) = 0 \]
Thus, \( \sin x = 0 \) or \( 5 \sin x - 4 = 0 \).

### Step 5: Find the solutions
1. \( \sin x = 0 \) gives \( x = 0 \) (not in the interval \( (0, \pi) \)).
2. \( 5 \sin x - 4 = 0 \) gives:
\[ \sin x = \frac{4}{5} \]
This gives \( x = \sin^{-1} \left( \frac{4}{5} \right) \) which is valid in the interval \( (0, \pi) \).

### Step 6: Check the endpoints
The function \( f(x) \) could also be non-differentiable at the endpoints of the interval, but since we are considering \( (0, \pi) \), we focus on the intersection point.

### Step 7: Identify non-differentiable points
The point of non-differentiability occurs at the intersection point found:
\[ x = \sin^{-1} \left( \frac{4}{5} \right) \]

### Conclusion
Thus, the set of points of non-differentiability for the function \( f(x) \) is:
\[ \left\{ \sin^{-1} \left( \frac{4}{5} \right) \right\} \]