91Ó°ÊÓ

We are given the function: $$f(x) = 3x^4 + 4x^3 - 12x^2$$.

To find the x-intercepts, set $$f(x) = 0$$ and solve for $$x$$: $$3x^4 + 4x^3 - 12x^2 = 0$$ Factor out $$x^2$$: $$x^2(3x^2 + 4x - 12) = 0$$ Now set $$x^2$$ and $$(3x^2 + 4x - 12)$$ equal to 0 separately and solve for $$x$$: - $$x^2 = 0 \Rightarrow x = 0$$ - $$3x^2 + 4x - 12 = 0$$ can be factored as $$(x + 2)(3x - 6) = 0$$, giving $$x = -2$$ and $$x = 2$$. So, the x-intercepts are $$(-2, 0), (0, 0),$$ and $$(2, 0)$$. For the y-intercept, set $$x = 0$$ and solve for $$y$$: $$f(0) = 3(0)^4 + 4(0)^3 - 12(0)^2 = 0$$ So, the y-intercept is $$(0, 0)$$.

To find the local extreme values, calculate the first derivative of the function, and then find the critical points by setting the derivative equal to 0: $$f'(x) = \frac{d}{dx}(3x^4 + 4x^3 - 12x^2) = 12x^3 + 12x^2 - 24x$$ Now, set $$f'(x) = 0$$: $$12x^3 + 12x^2 - 24x = 0 \Rightarrow x(12x^2 + 12x - 24) = 0$$ We already have one critical point, $$x = 0$$. For the other critical points, solve for $$x$$ in $$12x^2 + 12x - 24 = 0$$, which can be factored as $$(2x - 4)(6x + 6) = 0$$, giving $$x = 2$$ and $$x = -1$$. Now, find the second derivative to determine if these critical points are local maxima, local minima, or neither: $$f''(x) = \frac{d^2}{dx^2}(3x^4 + 4x^3 - 12x^2) = 36x^2 + 24x - 24$$ Evaluate $$f''(-1)$$, $$f''(0)$$, and $$f''(2)$$: - $$f''(-1) = 36 - 24 - 24 = -12 < 0$$, so $x = -1$$ results in a local maximum. - $$f''(0) = -24 < 0$$, so $x = 0$$ results in a local maximum. - $$f''(2) = 36(4) + 24(2) - 24 = 120 > 0$$, so $x = 2$$ results in a local minimum. So, the local extreme values are at the points $$(-1, f(-1))$$, $$(0, f(0))$$, and $$(2, f(2))$$.

To find the inflection points, set the second derivative equal to 0: $$36x^2 + 24x - 24 = 0$$ We can notice the discriminant of this quadratic equation is negative, which means there are no real roots, so there are no inflection points.

Now that we know the intercepts, local extreme values, and the absence of inflection points, we can graph the function using these points and the general characteristics of a quartic function: 1. Plot the intercepts at $$(-2, 0), (0, 0)$$, and $$(2, 0)$$. 2. Plot the local maximum points at $$(-1, f(-1))$$ and $$(0, f(0))$$. 3. Plot the local minimum point at $$(2, f(2))$$. 4. Since there are no inflection points, the graph will not change its concavity. When you plot the function with a graphing utility using these features, you should obtain a complete graph of the function $$f(x) = 3x^4 + 4x^3 - 12x^2$$.