To analyze the function \( f(x) = 3x^3 + 7x^2 + 3x - 1 \), we will go through each part step-by-step, providing calculations and reasoning for every question.
I. Function Analysis
a. Domain
The function \( f(x) = 3x^3 + 7x^2 + 3x - 1 \) is a polynomial function. Polynomial functions are defined for all real numbers, so: Domain: \( \mathbb{R} \) or \( (-\infty, \infty) \)
b. x-intercept, y-intercept
x-intercept: To find the x-intercepts, we set \( f(x) = 0 \): \[ 3x^3 + 7x^2 + 3x - 1 = 0 \] Given that \( (x + 1) \) is a factor, we will perform synthetic division or polynomial long division to divide \( f(x) \) by \( (x + 1) \).
Using synthetic division with root \( x = -1 \):
-1 | 3 7 3 -1
| -3 -4 1
-----------------
3 4 -1 0
This yields \( f(x) = (x + 1)(3x^2 + 4x - 1) \).
To find the remaining x-intercepts, we can set the quadratic factor equal to zero: \[ 3x^2 + 4x - 1 = 0 \] Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 3 \cdot (-1)}}{2 \cdot 3} = \frac{-4 \pm \sqrt{16 + 12}}{6} = \frac{-4 \pm \sqrt{28}}{6} = \frac{-4 \pm 2\sqrt{7}}{6} = \frac{-2 \pm \sqrt{7}}{3} \]
Thus, the x-intercepts are: \[ x = -1, \quad x = \frac{-2 + \sqrt{7}}{3}, \quad x = \frac{-2 - \sqrt{7}}{3} \]
y-intercept: To find the y-intercept, evaluate \( f(0) \): \[ f(0) = -1 \] So, the y-intercept is \( (0, -1) \).
c. Symmetry
A function is even if \( f(-x) = f(x) \) and odd if \( f(-x) = -f(x) \): Calculating \( f(-x) \): \[ f(-x) = 3(-x)^3 + 7(-x)^2 + 3(-x) - 1 = -3x^3 + 7x^2 - 3x - 1 \] This implies that \( f(-x) \neq f(x) \) and \( f(-x) \neq -f(x) \). Therefore, \( f(x) \) has no symmetry.
d. Asymptote
Since the function is a polynomial, it does not have asymptotes.
Asymptote: None.
e. Critical numbers, table of intervals, maxima, minima
Finding Critical Numbers: We need to find the first derivative: \[ f'(x) = 9x^2 + 14x + 3 \] Set the first derivative equal to zero: \[ 9x^2 + 14x + 3 = 0 \] Using the quadratic formula: \[ x = \frac{-14 \pm \sqrt{14^2 - 4 \cdot 9 \cdot 3}}{2 \cdot 9} = \frac{-14 \pm \sqrt{196 - 108}}{18} = \frac{-14 \pm \sqrt{88}}{18} = \frac{-14 \pm 2\sqrt{22}}{18} = \frac{-7 \pm \sqrt{22}}{9} \]
Critical Points:
- \( x_1 = \frac{-7 + \sqrt{22}}{9} \)
- \( x_2 = \frac{-7 - \sqrt{22}}{9} \)
Testing Intervals: We will construct a sign chart using the critical points. We can evaluate \( f'(x) \) in the intervals:
- \( (-\infty, \frac{-7 - \sqrt{22}}{9}) \)
- \( \left( \frac{-7 - \sqrt{22}}{9}, \frac{-7 + \sqrt{22}}{9} \right) \)
- \( \left( \frac{-7 + \sqrt{22}}{9}, \infty \right) \)
Evaluate \( f'(x) \): Choose test points in each of the intervals:
- \( x = -2 \) gives \( f'(-2) > 0 \) (increasing)
- \( x = -1 \) gives \( f'(-1) < 0 \) (decreasing)
- \( x = 0 \) gives \( f'(0) > 0 \) (increasing)
Based on this, we can state:
- Increasing: \( (-\infty, \frac{-7 - \sqrt{22}}{9}) \) and \( \left( \frac{-7 + \sqrt{22}}{9}, \infty \right) \)
- Decreasing: \( \left( \frac{-7 - \sqrt{22}}{9}, \frac{-7 + \sqrt{22}}{9} \right) \)
Maxima and Minima:
- Local maximum at \( x = \frac{-7 - \sqrt{22}}{9} \)
- Local minimum at \( x = \frac{-7 + \sqrt{22}}{9} \)
f. Points of inflection, concavity test results
Finding Points of Inflection: We need the second derivative: \[ f''(x) = 18x + 14 \] Setting \( f''(x) = 0 \): \[ 18x + 14 = 0 \Rightarrow x = -\frac{14}{18} = -\frac{7}{9} \]
Testing Intervals for Concavity:
- \( (-\infty, -\frac{7}{9}) \): Select \( x = -1 \) gives \( f''(-1) < 0 \) (concave down)
- \( (-\frac{7}{9}, \infty) \): Select \( x = 0 \) gives \( f''(0) > 0 \) (concave up)
Points of Inflection: There is a point of inflection at \( x = -\frac{7}{9} \).
II. Sketch of the Curve
-
Plotting the Points:
- x-intercepts: \( x = -1, \frac{-2 + \sqrt{7}}{3}, \frac{-2 - \sqrt{7}}{3} \)
- y-intercept: \( (0, -1) \)
- Local maximum at \( \frac{-7 - \sqrt{22}}{9} \) and local minimum at \( \frac{-7 + \sqrt{22}}{9} \)
- Point of inflection: \( x = -\frac{7}{9} \)
-
Behavior of the Function:
- Increase until \( \frac{-7 - \sqrt{22}}{9} \), decrease until \( \frac{-7 + \sqrt{22}}{9} \), and then increase again.
- Concave down before \( x = -\frac{7}{9} \) and concave up thereafter.
-
Drawing the Curve:
- Sketch a cubic curve that intersects at the identified x and y-intercepts, has the noted concavity changes, and peaks at the local maxima and minima.
The sketched curve should reflect these characteristics with a smooth continuous function showing increasing and decreasing intervals with points of inflection.
Note: The specific numerical values for the x-intercepts and critical points can further be estimated using a calculator for more accuracy when sketching.
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