The number of solution of `x^(3)+4x-1=0` in the interval `x in(-2,1)` is

Answer

To find the number of solutions of the equation \( x^3 + 4x - 1 = 0 \) in the interval \( x \in (-2, 1) \), we can follow these steps:

### Step 1: Define the function
Let \( f(x) = x^3 + 4x - 1 \). We want to find the number of roots of \( f(x) = 0 \) in the interval \( (-2, 1) \).

**Hint:** Define the function clearly to analyze its behavior.

### Step 2: Calculate the derivative
Differentiate the function to find critical points:
\[ f'(x) = 3x^2 + 4 \]

**Hint:** Finding the derivative helps us understand the increasing or decreasing nature of the function.

### Step 3: Analyze the derivative
Since \( f'(x) = 3x^2 + 4 \) is always positive (as \( 3x^2 \geq 0 \) and \( 4 > 0 \)), this means that \( f(x) \) is a strictly increasing function.

**Hint:** A strictly increasing function can cross the x-axis at most once.

### Step 4: Evaluate the function at the endpoints of the interval
Calculate \( f(-2) \) and \( f(1) \):
\[ f(-2) = (-2)^3 + 4(-2) - 1 = -8 - 8 - 1 = -17 \]
\[ f(1) = (1)^3 + 4(1) - 1 = 1 + 4 - 1 = 4 \]

**Hint:** Evaluating the function at the endpoints helps determine the sign changes.

### Step 5: Determine the sign change
Since \( f(-2) = -17 < 0 \) and \( f(1) = 4 > 0 \), and knowing that \( f(x) \) is continuous (as it is a polynomial), there must be at least one root in the interval \( (-2, 1) \).

**Hint:** A sign change between two points indicates the presence of a root.

### Step 6: Conclude the number of solutions
Since \( f(x) \) is strictly increasing and we found a sign change, there is exactly one solution to \( f(x) = 0 \) in the interval \( (-2, 1) \).

**Final Answer:** The number of solutions of the equation \( x^3 + 4x - 1 = 0 \) in the interval \( (-2, 1) \) is **1**.