Consider the function `f(theta) =4(sin^(2)theta + cos^(4)theta)`
What is the maximum value of the function `f(theta)`?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the maximum value of the function \( f(\theta) = 4(\sin^2 \theta + \cos^4 \theta) \), we will follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ f(\theta) = 4(\sin^2 \theta + \cos^4 \theta) \]

### Step 2: Use the Pythagorean identity
We know that \( \sin^2 \theta + \cos^2 \theta = 1 \). We can express \( \cos^4 \theta \) in terms of \( \sin^2 \theta \):
\[ \cos^4 \theta = (\cos^2 \theta)^2 = (1 - \sin^2 \theta)^2 \]
Thus, we can rewrite \( f(\theta) \):
\[ f(\theta) = 4(\sin^2 \theta + (1 - \sin^2 \theta)^2) \]

### Step 3: Expand the function
Now we expand \( (1 - \sin^2 \theta)^2 \):
\[ (1 - \sin^2 \theta)^2 = 1 - 2\sin^2 \theta + \sin^4 \theta \]
Substituting this back into \( f(\theta) \):
\[ f(\theta) = 4(\sin^2 \theta + 1 - 2\sin^2 \theta + \sin^4 \theta) = 4(1 - \sin^2 \theta + \sin^4 \theta) \]

### Step 4: Let \( x = \sin^2 \theta \)
Let \( x = \sin^2 \theta \). Then, \( \cos^2 \theta = 1 - x \) and we can rewrite the function as:
\[ f(x) = 4(1 - x + x^2) \]

### Step 5: Differentiate the function
To find the maximum value, we differentiate \( f(x) \):
\[ f'(x) = 4(-1 + 2x) \]
Setting the derivative to zero to find critical points:
\[ -1 + 2x = 0 \implies 2x = 1 \implies x = \frac{1}{2} \]

### Step 6: Find the second derivative
To confirm that this is a maximum, we check the second derivative:
\[ f''(x) = 4 \cdot 2 = 8 \]
Since \( f''(x) > 0 \), this indicates a minimum. Therefore, we need to check the endpoints of the interval \( x = 0 \) and \( x = 1 \).

### Step 7: Evaluate the function at critical points and endpoints
1. For \( x = 0 \):
\[ f(0) = 4(1 - 0 + 0^2) = 4 \]
2. For \( x = 1 \):
\[ f(1) = 4(1 - 1 + 1^2) = 4 \]
3. For \( x = \frac{1}{2} \):
\[ f\left(\frac{1}{2}\right) = 4\left(1 - \frac{1}{2} + \left(\frac{1}{2}\right)^2\right) = 4\left(1 - \frac{1}{2} + \frac{1}{4}\right) = 4\left(\frac{3}{4}\right) = 3 \]

### Conclusion
The maximum value of \( f(\theta) \) occurs at both \( x = 0 \) and \( x = 1 \), yielding:
\[ \text{Maximum value of } f(\theta) = 4 \]