If `A = sin^2x + cos^4 x`, then for all real x :

Options

A. `(13)/(16) le A le 1`
B. `1 le A le 2`
C. `3/4 le A le 13/16`
D. `3/4 le A le 1`

Answer

To find the range of \( A = \sin^2 x + \cos^4 x \) for all real \( x \), we can follow these steps:

### Step 1: Rewrite \( \cos^4 x \)
We know that \( \cos^2 x = 1 - \sin^2 x \). Therefore, we can express \( \cos^4 x \) in terms of \( \sin^2 x \):

\[ \cos^4 x = (\cos^2 x)^2 = (1 - \sin^2 x)^2 \]

### Step 2: Substitute \( \cos^4 x \) into \( A \)
Now substitute \( \cos^4 x \) into the expression for \( A \):

\[ A = \sin^2 x + (1 - \sin^2 x)^2 \]

### Step 3: Expand the expression
Next, we expand \( (1 - \sin^2 x)^2 \):

\[ (1 - \sin^2 x)^2 = 1 - 2\sin^2 x + \sin^4 x \]

Now substitute this back into the expression for \( A \):

\[ A = \sin^2 x + 1 - 2\sin^2 x + \sin^4 x \]

### Step 4: Combine like terms
Combine the terms in the expression:

\[ A = 1 - \sin^2 x + \sin^4 x \]

### Step 5: Let \( t = \sin^2 x \)
Let \( t = \sin^2 x \). Since \( \sin^2 x \) can take values between 0 and 1, we can rewrite \( A \):

\[ A = 1 - t + t^2 \]

### Step 6: Analyze the function \( A(t) \)
Now we need to analyze the quadratic function \( A(t) = t^2 - t + 1 \). To find the minimum value, we can complete the square or use calculus.

### Step 7: Find the vertex of the quadratic
The vertex of the quadratic \( A(t) = t^2 - t + 1 \) occurs at:

\[ t = -\frac{b}{2a} = -\frac{-1}{2 \cdot 1} = \frac{1}{2} \]

### Step 8: Calculate \( A \) at the vertex
Now substitute \( t = \frac{1}{2} \) back into \( A(t) \):

\[ A\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right) + 1 = \frac{1}{4} - \frac{1}{2} + 1 = \frac{1}{4} - \frac{2}{4} + \frac{4}{4} = \frac{3}{4} \]

### Step 9: Find the maximum value of \( A \)
Now we need to check the endpoints of the interval \( [0, 1] \):

1. For \( t = 0 \):
\[ A(0) = 0^2 - 0 + 1 = 1 \]

2. For \( t = 1 \):
\[ A(1) = 1^2 - 1 + 1 = 1 \]

### Conclusion
Thus, the minimum value of \( A \) is \( \frac{3}{4} \) and the maximum value is \( 1 \). Therefore, the range of \( A \) is:

\[ \frac{3}{4} \leq A \leq 1 \]

### Final Answer
\[ \frac{3}{4} \leq A \leq 1 \]