The number of solution of `sin^7x+cos^7x=1 , x in [0,4pi]`

Options

A. 7
B. 11
C. 9
D. 5

Answer

To find the number of solutions for the equation \( \sin^7 x + \cos^7 x = 1 \) in the interval \( x \in [0, 4\pi] \), we can follow these steps:

### Step 1: Analyze the Equation
We start with the equation:
\[ \sin^7 x + \cos^7 x = 1 \]
We know that both \( \sin^2 x \) and \( \cos^2 x \) are always less than or equal to 1. Therefore, \( \sin^7 x \) and \( \cos^7 x \) will also be less than or equal to \( \sin^2 x \) and \( \cos^2 x \) respectively.

### Step 2: Establish Upper Bound
From the properties of sine and cosine, we have:
\[ \sin^2 x + \cos^2 x = 1 \]
Thus:
\[ \sin^7 x \leq \sin^2 x \quad \text{and} \quad \cos^7 x \leq \cos^2 x \]
Adding these inequalities gives:
\[ \sin^7 x + \cos^7 x \leq \sin^2 x + \cos^2 x = 1 \]
This shows that \( \sin^7 x + \cos^7 x \) can never exceed 1.

### Step 3: Determine Conditions for Equality
For \( \sin^7 x + \cos^7 x = 1 \) to hold, either \( \sin^7 x = 1 \) and \( \cos^7 x = 0 \) or \( \sin^7 x = 0 \) and \( \cos^7 x = 1 \).

1. **Case 1**: \( \sin^7 x = 1 \)
- This implies \( \sin x = 1 \).
- The solutions for \( \sin x = 1 \) in the interval \( [0, 4\pi] \) are:
\[ x = \frac{\pi}{2} + 2k\pi, \quad k \in \mathbb{Z} \]
- For \( k = 0 \): \( x = \frac{\pi}{2} \)
- For \( k = 1 \): \( x = \frac{\pi}{2} + 2\pi = \frac{5\pi}{2} \)
- Thus, we have 2 solutions from this case.

2. **Case 2**: \( \cos^7 x = 1 \)
- This implies \( \cos x = 1 \).
- The solutions for \( \cos x = 1 \) in the interval \( [0, 4\pi] \) are:
\[ x = 2k\pi, \quad k \in \mathbb{Z} \]
- For \( k = 0 \): \( x = 0 \)
- For \( k = 1 \): \( x = 2\pi \)
- For \( k = 2 \): \( x = 4\pi \)
- Thus, we have 3 solutions from this case.

### Step 4: Count Total Solutions
Now, we combine the solutions from both cases:
- From Case 1: 2 solutions (\( \frac{\pi}{2}, \frac{5\pi}{2} \))
- From Case 2: 3 solutions (\( 0, 2\pi, 4\pi \))

Adding these gives us:
\[ 2 + 3 = 5 \text{ solutions} \]

### Conclusion
The total number of solutions to the equation \( \sin^7 x + \cos^7 x = 1 \) in the interval \( [0, 4\pi] \) is **5**.