The graph of the equation `y=x^(3)+5x+1`

Options

A. Does not intersect the x-axis
B. intersects the x-axis at one and only one point
C. intersects the x-axis at exactly three points
D. intersects the x-axis at more than three points.

Explanation

To determine how many times the graph of the equation \( y = x^3 + 5x + 1 \) intersects the x-axis, we need to find the values of \( x \) for which \( y = 0 \). This means we need to solve the equation:

\[ x^3 + 5x + 1 = 0 \]

### Step 1: Evaluate the function at specific points

Let's evaluate the function at a few points to understand its behavior:

1. **At \( x = -1 \)**:
\[ y = (-1)^3 + 5(-1) + 1 = -1 - 5 + 1 = -5 \]

2. **At \( x = 0 \)**:
\[ y = 0^3 + 5(0) + 1 = 1 \]

From these evaluations, we see that:
- At \( x = -1 \), \( y = -5 \) (negative)
- At \( x = 0 \), \( y = 1 \) (positive)

### Step 2: Analyze the sign change

Since the function changes from negative to positive between \( x = -1 \) and \( x = 0 \), by the Intermediate Value Theorem, there must be at least one root in the interval \( (-1, 0) \).

### Step 3: Evaluate the function at more points

To check if there are more roots, let's evaluate the function at other points:

3. **At \( x = 1 \)**:
\[ y = 1^3 + 5(1) + 1 = 1 + 5 + 1 = 7 \]

4. **At \( x = -2 \)**:
\[ y = (-2)^3 + 5(-2) + 1 = -8 - 10 + 1 = -17 \]

5. **At \( x = -3 \)**:
\[ y = (-3)^3 + 5(-3) + 1 = -27 - 15 + 1 = -41 \]

### Step 4: Analyze the behavior of the function

From the evaluations:
- At \( x = -2 \), \( y = -17 \) (negative)
- At \( x = -3 \), \( y = -41 \) (negative)
- At \( x = 1 \), \( y = 7 \) (positive)

The function is negative for \( x < -1 \) and becomes positive after \( x = 0 \). Since the function is a cubic polynomial, it will continue to increase without bound as \( x \) increases.

### Step 5: Conclusion

The function \( y = x^3 + 5x + 1 \) has:
- One root in the interval \( (-1, 0) \)
- No other roots for \( x > 0 \) since the function is increasing and positive.

Thus, the graph of the equation intersects the x-axis at exactly one point.

### Final Answer:
The graph intersects the x-axis at one and only one point.

---