The equation `x^(4) -7x + 2=0` has

Options

A. exactly two real and distinct solutions
B. has four real roots
C. no real root
D. all the four roots lie between 0 and 2

Answer

To determine the nature and number of roots of the equation \( x^4 - 7x + 2 = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We can express the equation in a form that allows us to analyze it better:
\[ x^4 = 7x - 2 \]

### Step 2: Define two functions
Let:
\[ f(x) = x^4 \]
\[ g(x) = 7x - 2 \]
We want to find the points where these two functions intersect, which corresponds to the roots of the equation.

### Step 3: Analyze the behavior of \( f(x) \) and \( g(x) \)
- The function \( f(x) = x^4 \) is a polynomial of degree 4, which is always non-negative and approaches infinity as \( x \) approaches ±∞. The graph is a U-shaped curve.
- The function \( g(x) = 7x - 2 \) is a linear function with a slope of 7. It crosses the y-axis at -2.

### Step 4: Find the derivative of \( f(x) \)
To understand the behavior of \( f(x) \), we can find its first derivative:
\[ f'(x) = 4x^3 \]
This derivative is zero at \( x = 0 \).

### Step 5: Analyze the critical points
- For \( x < 0 \), \( f'(x) < 0 \) (decreasing).
- For \( x = 0 \), \( f'(x) = 0 \) (local minimum).
- For \( x > 0 \), \( f'(x) > 0 \) (increasing).

This means \( f(x) \) has a local minimum at \( x = 0 \).

### Step 6: Evaluate \( f(0) \) and \( g(0) \)
Calculate the values of \( f \) and \( g \) at \( x = 0 \):
\[ f(0) = 0^4 = 0 \]
\[ g(0) = 7(0) - 2 = -2 \]
Since \( f(0) > g(0) \), the graph of \( f(x) \) is above the graph of \( g(x) \) at \( x = 0 \).

### Step 7: Find points of intersection
To find the points of intersection, we can analyze the behavior of both functions as \( x \) increases:
- As \( x \to +\infty \), \( f(x) \to +\infty \) and \( g(x) \to +\infty \).
- As \( x \to -\infty \), \( f(x) \to +\infty \) and \( g(x) \to -\infty \).

### Step 8: Use the Intermediate Value Theorem
Since \( f(x) \) is continuous and \( g(x) \) is linear, we can apply the Intermediate Value Theorem:
- There is at least one root in the interval \( (-\infty, 0) \) because \( f(x) \) starts high and goes low (crossing \( g(x) \)).
- There is at least one root in the interval \( (0, +\infty) \) because \( g(x) \) starts below \( f(x) \) at \( x = 0 \) and eventually goes above it.

### Step 9: Determine the number of roots
Since the functions intersect at two points, we conclude that there are two real roots.

### Conclusion
The equation \( x^4 - 7x + 2 = 0 \) has **two distinct real roots**.