Let `A=sin^8theta+cos^14theta, " then "A_("max")` is

Options

A. 1
B. `1/2`
C. `3/2`
D. None of these

Explanation

To find the maximum value of \( A = \sin^8 \theta + \cos^{14} \theta \), we can follow these steps:

### Step 1: Understand the Range of Sine and Cosine Functions
The sine and cosine functions have the following properties:
- \( -1 \leq \sin \theta \leq 1 \)
- \( -1 \leq \cos \theta \leq 1 \)

### Step 2: Analyze the Powers of Sine and Cosine
Since both sine and cosine are bounded between -1 and 1, we can deduce:
- \( 0 \leq \sin^2 \theta \leq 1 \) implies \( 0 \leq \sin^8 \theta \leq 1 \)
- \( 0 \leq \cos^2 \theta \leq 1 \) implies \( 0 \leq \cos^{14} \theta \leq 1 \)

### Step 3: Establish Upper Bounds
From the properties of powers:
- \( \sin^8 \theta \) is less than or equal to \( \sin^2 \theta \) because \( \sin^8 \theta \) is a higher power of a number between 0 and 1.
- Similarly, \( \cos^{14} \theta \) is less than or equal to \( \cos^2 \theta \).

### Step 4: Combine the Inequalities
Adding the two inequalities:
\[ \sin^8 \theta + \cos^{14} \theta \leq \sin^2 \theta + \cos^2 \theta \]
Since we know that \( \sin^2 \theta + \cos^2 \theta = 1 \), we can conclude:
\[ \sin^8 \theta + \cos^{14} \theta \leq 1 \]

### Step 5: Determine the Maximum Value
Thus, the maximum value of \( A \) is:
\[ A_{\text{max}} \leq 1 \]
To check if this maximum can actually be achieved, we can set \( \theta = 0 \):
- \( \sin(0) = 0 \) and \( \cos(0) = 1 \)
- Therefore, \( A = 0^8 + 1^{14} = 0 + 1 = 1 \)

### Conclusion
Thus, the maximum value of \( A \) is:
\[ \boxed{1} \]