Find maximum and minimum value of `cos^(2)theta+sin^(4)theta`.

Answer

To find the maximum and minimum values of the expression \( y = \cos^2 \theta + \sin^4 \theta \), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting \( \sin^4 \theta \) in terms of \( \sin^2 \theta \):
\[ y = \cos^2 \theta + \sin^4 \theta = \cos^2 \theta + (\sin^2 \theta)^2 \]
Let \( x = \sin^2 \theta \). Then, we can express \( \cos^2 \theta \) as:
\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - x \]
Substituting this into the equation gives:
\[ y = (1 - x) + x^2 \]
Thus, we have:
\[ y = x^2 - x + 1 \]

### Step 2: Find the critical points
To find the maximum and minimum values, we will differentiate \( y \) with respect to \( x \) and set the derivative to zero:
\[ \frac{dy}{dx} = 2x - 1 \]
Setting the derivative equal to zero:
\[ 2x - 1 = 0 \implies x = \frac{1}{2} \]

### Step 3: Evaluate the function at critical points and endpoints
Next, we evaluate \( y \) at \( x = \frac{1}{2} \) and at the endpoints \( x = 0 \) and \( x = 1 \):
1. For \( x = 0 \):
\[ y(0) = 0^2 - 0 + 1 = 1 \]
2. For \( x = 1 \):
\[ y(1) = 1^2 - 1 + 1 = 1 \]
3. For \( x = \frac{1}{2} \):
\[ y\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - \frac{1}{2} + 1 = \frac{1}{4} - \frac{1}{2} + 1 = \frac{1}{4} - \frac{2}{4} + \frac{4}{4} = \frac{3}{4} \]

### Step 4: Determine maximum and minimum values
Now we compare the values:
- \( y(0) = 1 \)
- \( y(1) = 1 \)
- \( y\left(\frac{1}{2}\right) = \frac{3}{4} \)

The maximum value of \( y \) is \( 1 \) and the minimum value of \( y \) is \( \frac{3}{4} \).

### Final Answer:
- Maximum value: \( 1 \)
- Minimum value: \( \frac{3}{4} \)