A function f (x)= max(sin x,cos x,1- cos x) is non-derivable for n values of $\mathrm{x}\in [0,2\mathrm{\pi }]$. Then the value of n is

The function f(x) is defined as the maximum of three functions: sin x, cos x, and 1 - cos x. Points of non-differentiability occur where there is a switch in which function is the maximum.

Step 2: Analyze Each Function

• sin x: Periodic with a maximum value of 1 at x = π/2 and a minimum value of -1 at x = 3π/2.
• cos x: Periodic with a maximum value of 1 at x = 0 and x = 2π, and a minimum value of -1 at x = π.
• 1 - cos x: Non-negative, with a maximum value of 2 at x = π and a minimum value of 0 at x = 0 and x = 2π.

Step 3: Identify Transitions in the Max Function

• sin x = cos x: Occurs at x = π/4 and x = 5π/4.
• sin x = 1 - cos x: Occurs at x = π/3 and x = 5π/3.
• cos x = 1 - cos x: Occurs at x = 0 and x = 2π.

Step 4: Points of Non-Differentiability

The function f(x) is non-differentiable at the points where the maximum value changes from one function to another:

• x = π/4: Switch between sin x and cos x.
• x = 5π/4: Switch between sin x and cos x.
• x = π/3: Switch between sin x and 1 - cos x.
• x = 5π/3: Switch between sin x and 1 - cos x.

Final Answer

There are 4 points of non-differentiability in f(x) over the interval [0, 2π]. The correct option is:

(d) 4