The function `y-a(1-cos x)` is maximum when `x` is equal to

Options

A. `pi`
B. `(pi)/2`
C. `-(pi)/2`
D. `-(pi)/6`

Explanation

To find the value of \( x \) at which the function \( y = a(1 - \cos x) \) is maximum, we can follow these steps:

### Step 1: Find the first derivative of the function.
The first derivative \( y' \) of the function \( y = a(1 - \cos x) \) is given by:
\[ y' = a \cdot \frac{d}{dx}(1 - \cos x) = a \cdot \sin x \]

### Step 2: Set the first derivative equal to zero to find critical points.
To find the critical points, we set the first derivative equal to zero:
\[ a \sin x = 0 \]
This gives us:
\[ \sin x = 0 \]
The solutions to this equation are:
\[ x = n\pi \quad \text{where } n \text{ is an integer} \]

### Step 3: Find the second derivative of the function.
Now, we find the second derivative \( y'' \):
\[ y'' = a \cdot \frac{d}{dx}(\sin x) = a \cos x \]

### Step 4: Determine the nature of the critical points using the second derivative.
To determine whether the critical points are maxima or minima, we evaluate the second derivative at the critical points:
\[ y'' = a \cos(n\pi) \]
Since \( \cos(n\pi) = (-1)^n \), we have:
\[ y'' = a (-1)^n \]

- For even \( n \) (e.g., \( n = 0, 2, 4, \ldots \)), \( y'' > 0 \) (indicating a local minimum).
- For odd \( n \) (e.g., \( n = 1, 3, 5, \ldots \)), \( y'' < 0 \) (indicating a local maximum).

### Step 5: Identify the maximum point.
The first odd integer is \( n = 1 \), which gives:
\[ x = \pi \]
Thus, the function \( y = a(1 - \cos x) \) is maximum when:
\[ x = \pi \]

### Final Answer:
The function \( y = a(1 - \cos x) \) is maximum when \( x = \pi \).

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