f(x)=maximum {2sinx,1-cosx} is not differentiable when x is equal to

Options

A. 1
B. -1
C. 0
D. `pi - cos^(-1) (3/5)`

Answer

To find the points where the function \( f(x) = \max\{2\sin x, 1 - \cos x\} \) is not differentiable, we need to analyze the two functions involved and determine where they intersect.

### Step 1: Identify the functions and their domains
We have two functions:
1. \( g_1(x) = 2\sin x \)
2. \( g_2(x) = 1 - \cos x \)

We will analyze these functions over the interval \( [0, \pi] \).

### Step 2: Find the intersection points
To find where \( f(x) \) is not differentiable, we need to find the points where \( g_1(x) = g_2(x) \).

Set the two functions equal to each other:
\[ 2\sin x = 1 - \cos x \]

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

### Step 4: Use trigonometric identities
We can use the identity \( \cos x = 1 - 2\sin^2\left(\frac{x}{2}\right) \) to rewrite the equation:
\[ 2\sin x + (1 - 2\sin^2\left(\frac{x}{2}\right)) = 1 \]
This simplifies to:
\[ 2\sin x + 1 - 2\sin^2\left(\frac{x}{2}\right) = 1 \]
Thus,
\[ 2\sin x = 2\sin^2\left(\frac{x}{2}\right) \]

### Step 5: Solve for \( \sin x \)
Using the double angle formula \( \sin x = 2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right) \):
\[ 2(2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right) = 2\sin^2\left(\frac{x}{2}\right) \]
Dividing both sides by 2 (assuming \( \sin\left(\frac{x}{2}\right) \neq 0 \)):
\[ 2\cos\left(\frac{x}{2}\right) = \sin\left(\frac{x}{2}\right) \]

### Step 6: Use the tangent function
This can be rewritten as:
\[ \tan\left(\frac{x}{2}\right) = 2 \]

### Step 7: Solve for \( x \)
Taking the arctangent gives:
\[ \frac{x}{2} = \tan^{-1}(2) \]
Thus,
\[ x = 2\tan^{-1}(2) \]

### Step 8: Find the value of \( x \)
To find the exact value of \( x \), we can also express it in terms of cosine:
\[ \cos x = 1 - \tan^2\left(\frac{x}{2}\right) / (1 + \tan^2\left(\frac{x}{2}\right)) \]
Substituting \( \tan\left(\frac{x}{2}\right) = 2 \):
\[ \cos x = 1 - \frac{4}{5} = -\frac{3}{5} \]

### Step 9: Find \( x \)
Thus, we have:
\[ x = \pi - \cos^{-1}\left(-\frac{3}{5}\right) \]
This is the point where \( f(x) \) is not differentiable.

### Final Answer
The function \( f(x) \) is not differentiable at:
\[ x = \pi - \cos^{-1}\left(\frac{3}{5}\right) \]