If `f(x) = |cos x| + |sin x|`, then `dy/dx` at `x = (2pi)/3` is equal to

Options

A. `(1-sqrt(3))/(2)`
B. 0
C. `(1)/(2)(sqrt(3)-1)`
D. none of these

Explanation

To find \( \frac{dy}{dx} \) at \( x = \frac{2\pi}{3} \) for the function \( f(x) = |\cos x| + |\sin x| \), we will follow these steps:

### Step 1: Determine the quadrant for \( x = \frac{2\pi}{3} \)
The angle \( \frac{2\pi}{3} \) radians corresponds to \( 120^\circ \), which is in the second quadrant. In this quadrant, the cosine function is negative, and the sine function is positive.

### Step 2: Rewrite the function using the properties of absolute values
Since \( \cos x \) is negative in the second quadrant, we have:
\[
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