The function `f(x)=x^6-x-1` cuts the x-axis

Options

A. at two points
B. at three points
C. at six points
D. not cut the x-axis

Answer

To determine how many times the function \( f(x) = x^6 - x - 1 \) cuts the x-axis, we need to find the real roots of the equation \( f(x) = 0 \). Here’s a step-by-step solution:

### Step 1: Set the function equal to zero
We start by setting the function equal to zero:
\[ f(x) = x^6 - x - 1 = 0 \]

### Step 2: Analyze the function
To understand how many times the function intersects the x-axis, we can analyze the behavior of the function. We will check the values of \( f(x) \) at certain points to see where it changes sign.

### Step 3: Evaluate \( f(x) \) at specific points
Let's evaluate \( f(x) \) at a few points:
- For \( x = 0 \):
\[ f(0) = 0^6 - 0 - 1 = -1 \]
- For \( x = 1 \):
\[ f(1) = 1^6 - 1 - 1 = -1 \]
- For \( x = 2 \):
\[ f(2) = 2^6 - 2 - 1 = 64 - 2 - 1 = 61 \]

### Step 4: Identify sign changes
From our evaluations:
- \( f(0) = -1 \) (negative)
- \( f(1) = -1 \) (negative)
- \( f(2) = 61 \) (positive)

Since \( f(0) \) and \( f(1) \) are both negative and \( f(2) \) is positive, there must be at least one root between \( x = 1 \) and \( x = 2 \).

### Step 5: Check for more roots
Next, we can check values between \( x = -1 \) and \( x = 0 \):
- For \( x = -1 \):
\[ f(-1) = (-1)^6 - (-1) - 1 = 1 + 1 - 1 = 1 \]
- For \( x = -0.5 \):
\[ f(-0.5) = (-0.5)^6 - (-0.5) - 1 = \frac{1}{64} + 0.5 - 1 \approx -0.484375 \]

This indicates that there is a root between \( x = -1 \) and \( x = -0.5 \) since \( f(-1) \) is positive and \( f(-0.5) \) is negative.

### Step 6: Conclusion
From our evaluations, we have identified:
1. One root between \( x = -1 \) and \( x = -0.5 \).
2. One root between \( x = 1 \) and \( x = 2 \).

Thus, the function \( f(x) = x^6 - x - 1 \) cuts the x-axis at **two points**.

### Final Answer
The function cuts the x-axis at **2 points**.

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