Find the points of local maxima and local minima, if any, of the following functions. Find also the local maximum and local minimum values :
`f(x)=cosx, 0ltxltpi`

Answer

To find the points of local maxima and minima for the function \( f(x) = \cos x \) in the interval \( 0 < x < \pi \), we will follow these steps:

### Step 1: Differentiate the function
We start by finding the first derivative of the function \( f(x) \).

\[ f'(x) = -\sin x \]

### Step 2: Set the first derivative to zero
To find the critical points, we set the first derivative equal to zero.

\[ -\sin x = 0 \implies \sin x = 0 \]

### Step 3: Solve for \( x \)
The solutions to \( \sin x = 0 \) are given by:

\[ x = n\pi \quad \text{for } n \in \mathbb{Z} \]

In the interval \( 0 < x < \pi \), the only solution is:

\[ x = 0 \quad \text{and} \quad x = \pi \]

However, since we are looking for points strictly within the interval \( (0, \pi) \), we note that there are no critical points in this interval.

### Step 4: Evaluate the endpoints
Since \( x = 0 \) and \( x = \pi \) are the endpoints of the interval, we need to evaluate the function at these points to find the local maxima and minima.

\[ f(0) = \cos(0) = 1 \]
\[ f(\pi) = \cos(\pi) = -1 \]

### Step 5: Determine local maxima and minima
- At \( x = 0 \), \( f(0) = 1 \) is a local maximum.
- At \( x = \pi \), \( f(\pi) = -1 \) is a local minimum.

### Conclusion
The function \( f(x) = \cos x \) has:
- A local maximum at \( x = 0 \) with a maximum value of \( 1 \).
- A local minimum at \( x = \pi \) with a minimum value of \( -1 \).

### Summary of Results
- Local maximum: \( x = 0 \), \( f(0) = 1 \)
- Local minimum: \( x = \pi \), \( f(\pi) = -1 \)

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