Find `max^(m)&min^(m)` value of `sin^(2)theta+costheta`

Answer

To find the maximum and minimum values of the function \( f(\theta) = \sin^2 \theta + \cos \theta \), we will follow these steps:

### Step 1: Rewrite the function
We start by rewriting \( \sin^2 \theta \) using the Pythagorean identity:
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
Thus, the function becomes:
\[ f(\theta) = 1 - \cos^2 \theta + \cos \theta = 1 - \cos^2 \theta + \cos \theta \]

### Step 2: Simplify the function
Now, we can combine the terms:
\[ f(\theta) = 1 + \cos \theta - \cos^2 \theta \]
This can be rearranged as:
\[ f(\theta) = -\cos^2 \theta + \cos \theta + 1 \]

### Step 3: Complete the square
To find the maximum and minimum values, we will complete the square for the quadratic expression in terms of \( \cos \theta \):
\[ f(\theta) = -(\cos^2 \theta - \cos \theta - 1) \]
Taking the coefficient of \( \cos \theta \) (which is -1), we take half of it, square it, and add/subtract it:
\[ f(\theta) = -\left(\cos^2 \theta - \cos \theta + \frac{1}{4} - \frac{1}{4} - 1\right) \]
This simplifies to:
\[ f(\theta) = -\left(\left(\cos \theta - \frac{1}{2}\right)^2 - \frac{5}{4}\right) \]
Thus, we have:
\[ f(\theta) = \frac{5}{4} - \left(\cos \theta - \frac{1}{2}\right)^2 \]

### Step 4: Determine maximum and minimum values
The term \(\left(\cos \theta - \frac{1}{2}\right)^2\) is always non-negative and achieves its minimum value of 0 when \(\cos \theta = \frac{1}{2}\). Therefore, the maximum value of \( f(\theta) \) occurs when:
\[ f(\theta)_{\text{max}} = \frac{5}{4} - 0 = \frac{5}{4} \]

For the minimum value, \(\left(\cos \theta - \frac{1}{2}\right)^2\) achieves its maximum value when \(\cos \theta\) is at its extremes, i.e., \(-1\) or \(1\).
- When \(\cos \theta = -1\):
\[ f(\theta)_{\text{min}} = \frac{5}{4} - \left(-1 - \frac{1}{2}\right)^2 = \frac{5}{4} - \left(-\frac{3}{2}\right)^2 = \frac{5}{4} - \frac{9}{4} = -1 \]

- When \(\cos \theta = 1\):
\[ f(\theta)_{\text{min}} = \frac{5}{4} - \left(1 - \frac{1}{2}\right)^2 = \frac{5}{4} - \left(\frac{1}{2}\right)^2 = \frac{5}{4} - \frac{1}{4} = 1 \]

Thus, the minimum value is:
\[ f(\theta)_{\text{min}} = -1 \]

### Conclusion
The maximum and minimum values of the function \( f(\theta) = \sin^2 \theta + \cos \theta \) are:
\[ \text{max}(f) = \frac{5}{4}, \quad \text{min}(f) = -1 \]