Consider the function `f(x)=max{|sinx|,|cosx|},AA"x"in[0,3pi].` if `lamda` is the number of points at which f(x) is non - differentiable , then value of `(lamda^3)/5` is

Answer

To solve the problem, we need to analyze the function \( f(x) = \max\{|\sin x|, |\cos x|\} \) over the interval \( [0, 3\pi] \) and determine the number of points where this function is non-differentiable.

### Step 1: Identify the points where \( |\sin x| \) and \( |\cos x| \) intersect
The function \( f(x) \) will be non-differentiable at points where \( |\sin x| = |\cos x| \). This occurs when:
\[ \tan x = 1 \quad \text{or} \quad \tan x = -1 \]
The solutions to \( \tan x = 1 \) in the interval \( [0, 3\pi] \) are:
\[ x = \frac{\pi}{4} + n\pi \quad (n = 0, 1, 2, 3, 4, 5) \]
Calculating these gives:
- For \( n = 0 \): \( x = \frac{\pi}{4} \)
- For \( n = 1 \): \( x = \frac{5\pi}{4} \)
- For \( n = 2 \): \( x = \frac{9\pi}{4} \)
- For \( n = 3 \): \( x = \frac{13\pi}{4} \)

The solutions to \( \tan x = -1 \) in the same interval are:
\[ x = \frac{3\pi}{4} + n\pi \quad (n = 0, 1, 2, 3, 4, 5) \]
Calculating these gives:
- For \( n = 0 \): \( x = \frac{3\pi}{4} \)
- For \( n = 1 \): \( x = \frac{7\pi}{4} \)
- For \( n = 2 \): \( x = \frac{11\pi}{4} \)
- For \( n = 3 \): \( x = \frac{15\pi}{4} \)

### Step 2: List all intersection points
Combining both sets of solutions, we find the following points:
1. \( x = \frac{\pi}{4} \)
2. \( x = \frac{3\pi}{4} \)
3. \( x = \frac{5\pi}{4} \)
4. \( x = \frac{7\pi}{4} \)
5. \( x = \frac{9\pi}{4} \)
6. \( x = \frac{11\pi}{4} \)
7. \( x = \frac{13\pi}{4} \)
8. \( x = \frac{15\pi}{4} \)

### Step 3: Count the non-differentiable points
From the above calculations, we have a total of 8 points where \( f(x) \) is non-differentiable.

### Step 4: Calculate \( \lambda^3 / 5 \)
Let \( \lambda = 8 \) (the number of non-differentiable points).
Now we calculate:
\[ \frac{\lambda^3}{5} = \frac{8^3}{5} = \frac{512}{5} = 102.4 \]

### Final Answer
Thus, the value of \( \frac{\lambda^3}{5} \) is \( \boxed{102.4} \).