Answer :
To determine which graph represents the function [tex]\( f(x) = x^4 + x^3 - 8x^2 - 12x \)[/tex], we need to analyze the function step by step.
1. Identify the degree and leading coefficient:
- The function [tex]\( f(x) = x^4 + x^3 - 8x^2 - 12x \)[/tex] is a quartic polynomial (degree 4) with a positive leading coefficient (1 for [tex]\( x^4 \)[/tex]). For large positive or large negative values of [tex]\( x \)[/tex], the function [tex]\( f(x) \)[/tex] will tend to [tex]\( +\infty \)[/tex], indicating the end behavior similar to an upward-opening parabola.
2. Finding critical points:
- To find the critical points, we need to calculate the derivative and solve for when it is equal to zero:
[tex]\[ f'(x) = 4x^3 + 3x^2 - 16x - 12 \][/tex]
Solving [tex]\( f'(x) = 0 \)[/tex]:
[tex]\[ 4x^3 + 3x^2 - 16x - 12 = 0 \][/tex]
Finding exact roots algebraically can be challenging without a computer, but we can approximate or use methods such as the Rational Root Theorem or numerical solvers.
3. Analyze the second derivative and concavity:
- To determine the concavity and inflection points, we compute the second derivative [tex]\( f''(x) \)[/tex]:
[tex]\[ f''(x) = 12x^2 + 6x - 16 \][/tex]
Solve [tex]\( f''(x) = 0 \)[/tex] to find potential inflection points:
[tex]\[ 12x^2 + 6x - 16 = 0 \][/tex]
This can be solved using the quadratic formula:
[tex]\[ x = \frac{-6 \pm \sqrt{6^2 - 4\times12\times(-16)}}{2\times12} = \frac{-6 \pm \sqrt{324}}{24} = \frac{-6 \pm 18}{24} \][/tex]
Thus, the inflection points are:
[tex]\[ x = \frac{12}{24} = \frac{1}{2}, \quad x = \frac{-24}{24} = -1 \][/tex]
4. Behavior near the roots:
- We can evaluate the function at several points to understand its behavior around these critical points and inflection points.
Substituting common points like [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0 \][/tex]
Evaluate at [tex]\( x = \pm1, \pm2 \)[/tex], etc., to understand slope and curvature changes.
Given this analysis, generally, the graph of [tex]\( f(x) = x^4 + x^3 - 8x^2 - 12x \)[/tex] would:
- Cross the x-axis at points where the function value is zero (roots of the polynomial).
- Have a turning point (local maximum or minimum) derived from solving [tex]\( f'(x) = 0 \)[/tex].
- Show changes in concavity as determined by the roots of [tex]\( f''(x) \)[/tex].
Finally, the graph of this polynomial should show "W"-shaped behavior often seen in quartic functions with a positive leading coefficient.
Observing these characteristics will help identify the correct graph among the choices provided. Look for a graph that has:
- 4th degree polynomial behavior (upward tails on both ends).
- Smooth continuous nature with multiple turning points.
By considering these specific features, you can match the suitable graph to the function [tex]\( f(x) = x^4 + x^3 - 8x^2 - 12x \)[/tex].
1. Identify the degree and leading coefficient:
- The function [tex]\( f(x) = x^4 + x^3 - 8x^2 - 12x \)[/tex] is a quartic polynomial (degree 4) with a positive leading coefficient (1 for [tex]\( x^4 \)[/tex]). For large positive or large negative values of [tex]\( x \)[/tex], the function [tex]\( f(x) \)[/tex] will tend to [tex]\( +\infty \)[/tex], indicating the end behavior similar to an upward-opening parabola.
2. Finding critical points:
- To find the critical points, we need to calculate the derivative and solve for when it is equal to zero:
[tex]\[ f'(x) = 4x^3 + 3x^2 - 16x - 12 \][/tex]
Solving [tex]\( f'(x) = 0 \)[/tex]:
[tex]\[ 4x^3 + 3x^2 - 16x - 12 = 0 \][/tex]
Finding exact roots algebraically can be challenging without a computer, but we can approximate or use methods such as the Rational Root Theorem or numerical solvers.
3. Analyze the second derivative and concavity:
- To determine the concavity and inflection points, we compute the second derivative [tex]\( f''(x) \)[/tex]:
[tex]\[ f''(x) = 12x^2 + 6x - 16 \][/tex]
Solve [tex]\( f''(x) = 0 \)[/tex] to find potential inflection points:
[tex]\[ 12x^2 + 6x - 16 = 0 \][/tex]
This can be solved using the quadratic formula:
[tex]\[ x = \frac{-6 \pm \sqrt{6^2 - 4\times12\times(-16)}}{2\times12} = \frac{-6 \pm \sqrt{324}}{24} = \frac{-6 \pm 18}{24} \][/tex]
Thus, the inflection points are:
[tex]\[ x = \frac{12}{24} = \frac{1}{2}, \quad x = \frac{-24}{24} = -1 \][/tex]
4. Behavior near the roots:
- We can evaluate the function at several points to understand its behavior around these critical points and inflection points.
Substituting common points like [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0 \][/tex]
Evaluate at [tex]\( x = \pm1, \pm2 \)[/tex], etc., to understand slope and curvature changes.
Given this analysis, generally, the graph of [tex]\( f(x) = x^4 + x^3 - 8x^2 - 12x \)[/tex] would:
- Cross the x-axis at points where the function value is zero (roots of the polynomial).
- Have a turning point (local maximum or minimum) derived from solving [tex]\( f'(x) = 0 \)[/tex].
- Show changes in concavity as determined by the roots of [tex]\( f''(x) \)[/tex].
Finally, the graph of this polynomial should show "W"-shaped behavior often seen in quartic functions with a positive leading coefficient.
Observing these characteristics will help identify the correct graph among the choices provided. Look for a graph that has:
- 4th degree polynomial behavior (upward tails on both ends).
- Smooth continuous nature with multiple turning points.
By considering these specific features, you can match the suitable graph to the function [tex]\( f(x) = x^4 + x^3 - 8x^2 - 12x \)[/tex].
