Find the points of local maxima/minima of following functions
(i) `f (x) = x + (1)/(x)`
(ii) f(x) = cosec x
Hence find maxima and minima values of f(x).

Answer

To find the points of local maxima and minima for the given functions, we will follow these steps:

### (i) For the function \( f(x) = x + \frac{1}{x} \):

1. **Find the derivative**:
\[ f'(x) = 1 - \frac{1}{x^2} \]

2. **Set the derivative to zero to find critical points**:
\[ 1 - \frac{1}{x^2} = 0 \implies \frac{1}{x^2} = 1 \implies x^2 = 1 \implies x = 1 \text{ or } x = -1 \]

3. **Determine the second derivative**:
\[ f''(x) = \frac{2}{x^3} \]

4. **Evaluate the second derivative at the critical points**:
- For \( x = 1 \):
\[ f''(1) = \frac{2}{1^3} = 2 \quad (\text{positive, so } x = 1 \text{ is a local minimum}) \]
- For \( x = -1 \):
\[ f''(-1) = \frac{2}{(-1)^3} = -2 \quad (\text{negative, so } x = -1 \text{ is a local maximum}) \]

5. **Find the function values at the critical points**:
- At \( x = 1 \):
\[ f(1) = 1 + \frac{1}{1} = 2 \]
- At \( x = -1 \):
\[ f(-1) = -1 + \frac{1}{-1} = -2 \]

### Summary for (i):
- Local minimum at \( x = 1 \) with value \( f(1) = 2 \).
- Local maximum at \( x = -1 \) with value \( f(-1) = -2 \).

### (ii) For the function \( f(x) = \csc x \):

1. **Find the derivative**:
\[ f'(x) = -\csc x \cot x \]

2. **Set the derivative to zero to find critical points**:
\[ -\csc x \cot x = 0 \implies \cot x = 0 \implies x = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]

3. **Determine the second derivative**:
\[ f''(x) = \csc x (\cot^2 x - \csc^2 x) \]

4. **Evaluate the second derivative at critical points**:
- For \( x = \frac{\pi}{2} \):
\[ f''\left(\frac{\pi}{2}\right) = \csc\left(\frac{\pi}{2}\right) \left(\cot^2\left(\frac{\pi}{2}\right) - \csc^2\left(\frac{\pi}{2}\right)\right) = 1(0 - 1) = -1 \quad (\text{negative, so } x = \frac{\pi}{2} \text{ is a local maximum}) \]

5. **Find the function value at the critical point**:
- At \( x = \frac{\pi}{2} \):
\[ f\left(\frac{\pi}{2}\right) = \csc\left(\frac{\pi}{2}\right) = 1 \]

### Summary for (ii):
- Local maximum at \( x = \frac{\pi}{2} \) with value \( f\left(\frac{\pi}{2}\right) = 1 \).