Find maximum value of `sin^(8)theta+cos^(14)theta`.

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ y = \sin^8 \theta + \cos^{14} \theta \]
We can express \( \cos^{14} \theta \) in terms of \( \sin^2 \theta \):
\[ y = \sin^8 \theta + (\cos^2 \theta)^7 \]
Let \( x = \sin^2 \theta \). Then, we have \( \cos^2 \theta = 1 - x \). Thus, we can rewrite the expression as:
\[ y = x^4 + (1 - x)^7 \]

### Step 2: Differentiate the expression
Next, we need to find the critical points of the function \( y \). We differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 4x^3 - 7(1 - x)^6 \]

### Step 3: Set the derivative to zero
To find the maximum, we set the derivative equal to zero:
\[ 4x^3 - 7(1 - x)^6 = 0 \]
This gives us:
\[ 4x^3 = 7(1 - x)^6 \]

### Step 4: Solve for \( x \)
This equation can be solved numerically or graphically, but we can also evaluate the function at the endpoints \( x = 0 \) and \( x = 1 \) to find the maximum value.

### Step 5: Evaluate at endpoints
1. When \( x = 0 \):
\[ y = 0^4 + (1 - 0)^7 = 0 + 1 = 1 \]
2. When \( x = 1 \):
\[ y = 1^4 + (1 - 1)^7 = 1 + 0 = 1 \]

### Step 6: Check the value at \( x = \frac{4}{11} \) (approx)
We can also evaluate \( y \) at \( x = \frac{4}{11} \) (found from solving the derivative):
\[ y = \left(\frac{4}{11}\right)^4 + \left(1 - \frac{4}{11}\right)^7 \]
Calculating this gives a value less than 1.

### Conclusion
The maximum value of \( y = \sin^8 \theta + \cos^{14} \theta \) occurs at both endpoints \( x = 0 \) and \( x = 1 \), yielding:
\[ \text{Maximum value} = 1 \]