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

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the minimum value of the function \( f(\theta) = 4(\sin^2 \theta + \cos^4 \theta) \), we can 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 identity for \( \cos^4 \theta \)
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 \( f(\theta) \) as:
\[ f(\theta) = 4(\sin^2 \theta + (1 - \sin^2 \theta)^2) \]

### Step 3: Expand the expression
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 \( f(\theta) \):
\[ f(\theta) = 4(\sin^2 \theta + 1 - 2\sin^2 \theta + \sin^4 \theta) \]
This simplifies to:
\[ f(\theta) = 4(1 - \sin^2 \theta + \sin^4 \theta) \]

### Step 4: Let \( x = \sin^2 \theta \)
Let \( x = \sin^2 \theta \), where \( 0 \leq x \leq 1 \). Then we have:
\[ f(x) = 4(1 - x + x^2) \]

### Step 5: Differentiate to find critical points
Now, 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: Evaluate \( f(x) \) at the critical point
Now we evaluate \( f(x) \) at \( x = \frac{1}{2} \):
\[ f\left(\frac{1}{2}\right) = 4\left(1 - \frac{1}{2} + \left(\frac{1}{2}\right)^2\right) = 4\left(1 - \frac{1}{2} + \frac{1}{4}\right) \]
Calculating inside the parentheses:
\[ 1 - \frac{1}{2} + \frac{1}{4} = \frac{4}{4} - \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \]
Thus,
\[ f\left(\frac{1}{2}\right) = 4 \times \frac{3}{4} = 3 \]

### Step 7: Check the endpoints
We also need to check the endpoints \( x = 0 \) and \( x = 1 \):
- For \( x = 0 \):
\[ f(0) = 4(1 - 0 + 0^2) = 4 \]
- For \( x = 1 \):
\[ f(1) = 4(1 - 1 + 1^2) = 4 \]

### Conclusion
The minimum value of \( f(\theta) \) occurs at \( x = \frac{1}{2} \) and is:
\[ \text{Minimum value of } f(\theta) = 3 \]