The minimum value of `sin^2 theta + cos^4 theta` is

Options

A. `1/sqrt2`
B. `3/5`
C. `3/4`
D. `2/3`

Answer

To find the minimum value of the expression \( \sin^2 \theta + \cos^4 \theta \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ y = \sin^2 \theta + \cos^4 \theta \]

### Step 2: Use the Pythagorean identity
Recall that \( \sin^2 \theta + \cos^2 \theta = 1 \). We can express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
Let \( x = \cos^2 \theta \). Then, we can rewrite \( y \) as:
\[ y = (1 - x) + x^2 \]
This simplifies to:
\[ y = x^2 - x + 1 \]

### Step 3: Find the derivative
To find the minimum value, we can take the derivative of \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 2x - 1 \]
Setting the derivative equal to zero to find critical points:
\[ 2x - 1 = 0 \implies x = \frac{1}{2} \]

### Step 4: Determine the second derivative
To confirm that this critical point is a minimum, we can check the second derivative:
\[ \frac{d^2y}{dx^2} = 2 \]
Since \( \frac{d^2y}{dx^2} > 0 \), the function is concave up, indicating a minimum at \( x = \frac{1}{2} \).

### Step 5: Calculate the minimum value
Substituting \( x = \frac{1}{2} \) back into the expression for \( y \):
\[ y = \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right) + 1 = \frac{1}{4} - \frac{1}{2} + 1 \]
Finding a common denominator (4):
\[ y = \frac{1}{4} - \frac{2}{4} + \frac{4}{4} = \frac{1 - 2 + 4}{4} = \frac{3}{4} \]

### Conclusion
Thus, the minimum value of \( \sin^2 \theta + \cos^4 \theta \) is:
\[ \boxed{\frac{3}{4}} \]