The maximum value of `sin^8 theta + cos^14 theta` for all real values of `theta` is s

Options

A. 1
B. `sqrt2`
C. `1/sqrt2`
D. 0

Answer

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

### Step 1: Use the Pythagorean Identity
We know from trigonometry that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Let \( x = \sin^2 \theta \). Then, we can express \( \cos^2 \theta \) as:
\[ \cos^2 \theta = 1 - x \]
Thus, we can rewrite the expression \( \sin^8 \theta + \cos^{14} \theta \) in terms of \( x \):
\[ \sin^8 \theta = x^4 \quad \text{and} \quad \cos^{14} \theta = (1 - x)^7 \]
So, the expression becomes:
\[ f(x) = x^4 + (1 - x)^7 \]

### Step 2: Find the Derivative
To find the maximum value, we need to take the derivative of \( f(x) \) and set it to zero:
\[ f'(x) = 4x^3 - 7(1 - x)^6 \]
Setting the derivative equal to zero gives us:
\[ 4x^3 = 7(1 - x)^6 \]

### Step 3: Solve for Critical Points
This equation is complex, so we can analyze the behavior of \( f(x) \) at the endpoints and critical points. The endpoints are \( x = 0 \) and \( x = 1 \).

### Step 4: Evaluate at Endpoints
1. **At \( x = 0 \)**:
\[ f(0) = 0^4 + (1 - 0)^7 = 0 + 1 = 1 \]

2. **At \( x = 1 \)**:
\[ f(1) = 1^4 + (1 - 1)^7 = 1 + 0 = 1 \]

### Step 5: Check the Maximum Value
Since \( f(x) \) is continuous on the interval \([0, 1]\) and we have evaluated it at the endpoints, we can conclude that the maximum value occurs at both endpoints:
\[ \text{Maximum value} = 1 \]

### Conclusion
Thus, the maximum value of \( \sin^8 \theta + \cos^{14} \theta \) for all real values of \( \theta \) is:
\[ \boxed{1} \]