Certainly! Let's go through the detailed, step-by-step solution for the given function [tex]\( f(x) = x^3 - 3x + 2 \)[/tex] over the interval [tex]\([-2, 2]\)[/tex].
### Step 1: Define the Function
The function is given by:
[tex]\[ f(x) = x^3 - 3x + 2 \][/tex]
### Step 2: Determine the Interval
We are considering the interval [tex]\([-2, 2]\)[/tex].
### Step 3: Generate Points within the Interval
We'll consider a set of points (both [tex]\( x \)[/tex] and corresponding [tex]\( f(x) \)[/tex] values) within the interval [tex]\([-2, 2]\)[/tex] for better comprehension and visualization of the function.
Let’s list the x-values within the interval [tex]\([-2, 2]\)[/tex]:
[tex]\[ x = [-2.0, -1.989, -1.98, -1.97, ..., 1.98, 1.99, 2.0] \][/tex]
### Step 4: Evaluate [tex]\( f(x) \)[/tex] at these Points
We evaluate the function [tex]\( f(x) \)[/tex] at the points within [tex]\([-2, 2]\)[/tex]. Here are a few calculated values of [tex]\( f(x) \)[/tex]:
[tex]\[ y = [0.0, 0.0896, 0.178, 0.265, ..., 0.178, 0.0896, 0.0] \][/tex]
### Step 5: Calculate the First Derivative of [tex]\( f(x) \)[/tex]
The first derivative of the function [tex]\( f(x) \)[/tex] is:
[tex]\[ f'(x) = \frac{d}{dx}(x^3 - 3x + 2) = 3x^2 - 3 \][/tex]
### Step 6: Calculate the Second Derivative of [tex]\( f(x) \)[/tex]
The second derivative of the function [tex]\( f(x) \)[/tex] is:
[tex]\[ f''(x) = \frac{d}{dx}(3x^2 - 3) = 6x \][/tex]
### Summary of Results
- Function: [tex]\( f(x) = x^3 - 3x + 2 \)[/tex]
- First Derivative: [tex]\( f'(x) = 3x^2 - 3 \)[/tex]
- Second Derivative: [tex]\( f''(x) = 6x \)[/tex]
- X-values (sampled within the interval [tex]\([-2, 2]\)[/tex]):
[tex]\[ x = [-2.0, -1.989, -1.98, ..., 1.98, 1.99, 2.0] \][/tex]
- Corresponding [tex]\( f(x) \)[/tex] values:
[tex]\[ y = [0.0, 0.0896, 0.178, ..., 0.178, 0.0896, 0.0] \][/tex]
By examining the [tex]\( f(x) \)[/tex] values and the computed first and second derivatives, we can gain a comprehensive understanding of the behavior of the function within the interval [tex]\([-2, 2]\)[/tex].
