Answer :
Certainly! Let’s explore and analyze the function \( F(x) = x^3 + 3x^2 - 8 \) in a detailed, step-by-step manner.
### Step 1: Identify the Function
The given function is:
[tex]\[ F(x) = x^3 + 3x^2 - 8 \][/tex]
### Step 2: Understanding the Function
This is a polynomial function of degree 3, which means it is a cubic function. The general form of a cubic function is:
[tex]\[ y = ax^3 + bx^2 + cx + d \][/tex]
In this case:
- \( a = 1 \)
- \( b = 3 \)
- \( c = 0 \)
- \( d = -8 \)
### Step 3: Finding the Roots of the Function
To find the roots (or zeros) of the function, we need to solve the equation:
[tex]\[ x^3 + 3x^2 - 8 = 0 \][/tex]
Finding the roots of a cubic equation analytically can be complex, but it can be solved using methods like factoring, the Rational Root Theorem, or numerical techniques.
### Step 4: First Derivative – Analyzing the Turning Points
To find the local maxima and minima, we take the first derivative of the function:
[tex]\[ F'(x) = \frac{d}{dx} (x^3 + 3x^2 - 8) \][/tex]
Calculating the derivative:
[tex]\[ F'(x) = 3x^2 + 6x \][/tex]
Set the derivative equal to zero to find critical points:
[tex]\[ 3x^2 + 6x = 0 \][/tex]
[tex]\[ 3x(x + 2) = 0 \][/tex]
Solving for \( x \):
[tex]\[ x = 0 \][/tex]
[tex]\[ x = -2 \][/tex]
### Step 5: Second Derivative – Analyzing Concavity
To determine the nature of these critical points, we take the second derivative of the function:
[tex]\[ F''(x) = \frac{d}{dx} (3x^2 + 6x) \][/tex]
Calculating the second derivative:
[tex]\[ F''(x) = 6x + 6 \][/tex]
Evaluate the second derivative at the critical points:
- At \( x = 0 \):
[tex]\[ F''(0) = 6(0) + 6 = 6 \][/tex]
Since \( F''(0) > 0 \), the function has a local minimum at \( x = 0 \).
- At \( x = -2 \):
[tex]\[ F''(-2) = 6(-2) + 6 = -12 + 6 = -6 \][/tex]
Since \( F''(-2) < 0 \), the function has a local maximum at \( x = -2 \).
### Step 6: Behavior at Infinity
As \( x \to \pm\infty \), the \( x^3 \) term will dominate, and the function will behave similarly to \( x^3 \):
[tex]\[ \lim_{x \to \infty} F(x) = \infty \][/tex]
[tex]\[ \lim_{x \to -\infty} F(x) = -\infty \][/tex]
### Step 7: Summary of Findings
- The function \( F(x) = x^3 + 3x^2 - 8 \) is a cubic polynomial.
- It has roots where \( F(x) = 0 \).
- It has a local maximum at \( x = -2 \) and a local minimum at \( x = 0 \).
- The function tends to \(\infty\) as \( x \to \infty \) and \(-\infty\) as \( x \to -\infty \).
Thus, the detailed analysis of the function [tex]\( F(x) = x^3 + 3x^2 - 8 \)[/tex] provides insights into its behavior and critical points.
### Step 1: Identify the Function
The given function is:
[tex]\[ F(x) = x^3 + 3x^2 - 8 \][/tex]
### Step 2: Understanding the Function
This is a polynomial function of degree 3, which means it is a cubic function. The general form of a cubic function is:
[tex]\[ y = ax^3 + bx^2 + cx + d \][/tex]
In this case:
- \( a = 1 \)
- \( b = 3 \)
- \( c = 0 \)
- \( d = -8 \)
### Step 3: Finding the Roots of the Function
To find the roots (or zeros) of the function, we need to solve the equation:
[tex]\[ x^3 + 3x^2 - 8 = 0 \][/tex]
Finding the roots of a cubic equation analytically can be complex, but it can be solved using methods like factoring, the Rational Root Theorem, or numerical techniques.
### Step 4: First Derivative – Analyzing the Turning Points
To find the local maxima and minima, we take the first derivative of the function:
[tex]\[ F'(x) = \frac{d}{dx} (x^3 + 3x^2 - 8) \][/tex]
Calculating the derivative:
[tex]\[ F'(x) = 3x^2 + 6x \][/tex]
Set the derivative equal to zero to find critical points:
[tex]\[ 3x^2 + 6x = 0 \][/tex]
[tex]\[ 3x(x + 2) = 0 \][/tex]
Solving for \( x \):
[tex]\[ x = 0 \][/tex]
[tex]\[ x = -2 \][/tex]
### Step 5: Second Derivative – Analyzing Concavity
To determine the nature of these critical points, we take the second derivative of the function:
[tex]\[ F''(x) = \frac{d}{dx} (3x^2 + 6x) \][/tex]
Calculating the second derivative:
[tex]\[ F''(x) = 6x + 6 \][/tex]
Evaluate the second derivative at the critical points:
- At \( x = 0 \):
[tex]\[ F''(0) = 6(0) + 6 = 6 \][/tex]
Since \( F''(0) > 0 \), the function has a local minimum at \( x = 0 \).
- At \( x = -2 \):
[tex]\[ F''(-2) = 6(-2) + 6 = -12 + 6 = -6 \][/tex]
Since \( F''(-2) < 0 \), the function has a local maximum at \( x = -2 \).
### Step 6: Behavior at Infinity
As \( x \to \pm\infty \), the \( x^3 \) term will dominate, and the function will behave similarly to \( x^3 \):
[tex]\[ \lim_{x \to \infty} F(x) = \infty \][/tex]
[tex]\[ \lim_{x \to -\infty} F(x) = -\infty \][/tex]
### Step 7: Summary of Findings
- The function \( F(x) = x^3 + 3x^2 - 8 \) is a cubic polynomial.
- It has roots where \( F(x) = 0 \).
- It has a local maximum at \( x = -2 \) and a local minimum at \( x = 0 \).
- The function tends to \(\infty\) as \( x \to \infty \) and \(-\infty\) as \( x \to -\infty \).
Thus, the detailed analysis of the function [tex]\( F(x) = x^3 + 3x^2 - 8 \)[/tex] provides insights into its behavior and critical points.
