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 change its expression at points where \( |\sin x| = |\cos x| \). This occurs when:
\[ \sin x = \cos x \quad \text{or} \quad \sin x = -\cos x \]
These equations can be solved as follows:

1. For \( \sin x = \cos x \):
\[ \tan x = 1 \implies x = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z} \]

2. For \( \sin x = -\cos x \):
\[ \tan x = -1 \implies x = \frac{3\pi}{4} + n\pi, \quad n \in \mathbb{Z} \]

### Step 2: Find the specific points in the interval \( [0, 3\pi] \)
Now we will find the specific points in the interval \( [0, 3\pi] \):

1. From \( \tan x = 1 \):
- \( x = \frac{\pi}{4} \)
- \( x = \frac{5\pi}{4} \)
- \( x = \frac{9\pi}{4} \)

2. From \( \tan x = -1 \):
- \( x = \frac{3\pi}{4} \)
- \( x = \frac{7\pi}{4} \)
- \( x = \frac{11\pi}{4} \)

### Step 3: List all intersection points
The intersection points in the interval \( [0, 3\pi] \) are:
- \( \frac{\pi}{4} \)
- \( \frac{3\pi}{4} \)
- \( \frac{5\pi}{4} \)
- \( \frac{7\pi}{4} \)
- \( \frac{9\pi}{4} \)
- \( \frac{11\pi}{4} \)

### Step 4: Count the non-differentiable points
The function \( f(x) \) is non-differentiable at each of these points because the maximum function can switch from one piece to another at these intersections. Thus, we have a total of 6 non-differentiable points.

### Step 5: Calculate \( \lambda^3 / 5 \)
Let \( \lambda \) be the number of non-differentiable points:
\[ \lambda = 6 \]
Now we calculate:
\[ \frac{\lambda^3}{5} = \frac{6^3}{5} = \frac{216}{5} = 43.2 \]

### Final Answer
Thus, the value of \( \frac{\lambda^3}{5} \) is \( \boxed{43.2} \).