Show that `f(x)=sinx(1+cosx)` is maximum at `x=pi/3` in the interval `[0,\ pi]` .

Explanation

To show that the function \( f(x) = \sin x (1 + \cos x) \) has a maximum at \( x = \frac{\pi}{3} \) in the interval \([0, \pi]\), we will follow these steps:

### Step 1: Find the first derivative \( f'(x) \)

Using the product rule, we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(\sin x) \cdot (1 + \cos x) + \sin x \cdot \frac{d}{dx}(1 + \cos x) \]
...