1. The problem is to analyze the behavior of the function $$f(x) = -0.5x^4 + 1.2x^2 - 0.9x - 1$$ as $$x \to \infty$$ and $$x \to -\infty$$ and understand the shape of its graph. 2. The leading term of the polynomial, $$-0.5x^4$$, dominates the behavior for very large positive or negative values of $$x$$ because it has the highest power. 3. Since the coefficient of $$x^4$$ is negative ($$-0.5$$), as $$x \to \infty$$, $$f(x) \to -\infty$$ and as $$x \to -\infty$$, $$f(x) \to -\infty$$. 4. This means the ends of the graph go downwards on both sides. 5. The function $$g(x)$$ mentioned tends to $$+\infty$$, but since it is not defined here, we focus on $$f(x)$$. 6. The graph shape described as a folded sheet-like shape at the bottom-right corner suggests a local minimum or a fold near that region. 7. To find critical points, we differentiate $$f(x)$$: $$f'(x) = -0.5 \times 4x^3 + 1.2 \times 2x - 0.9 = -2x^3 + 2.4x - 0.9$$ 8. Set $$f'(x) = 0$$ to find critical points: $$-2x^3 + 2.4x - 0.9 = 0$$ 9. Divide both sides by $$-0.3$$ to simplify: $$\cancel{-2}x^3 + \cancel{2.4}x - \cancel{0.9} = 0 \Rightarrow \frac{-2}{-0.3}x^3 + \frac{2.4}{-0.3}x - \frac{0.9}{-0.3} = 0$$ Actually, better to divide by $$-0.3$$: $$\frac{-2}{-0.3}x^3 + \frac{2.4}{-0.3}x - \frac{0.9}{-0.3} = 0$$ Calculate: $$\frac{-2}{-0.3} = \frac{2}{0.3} = \frac{20}{3}$$ $$\frac{2.4}{-0.3} = -8$$ $$\frac{0.9}{-0.3} = -3$$ So: $$\frac{20}{3}x^3 - 8x + 3 = 0$$ 10. This cubic can be solved numerically or graphically to find critical points. 11. The behavior and shape of the graph near these points explain the folded sheet-like shape. Final answer: - As $$x \to \infty$$, $$f(x) \to -\infty$$. - As $$x \to -\infty$$, $$f(x) \to -\infty$$. - The graph has local extrema found by solving $$-2x^3 + 2.4x - 0.9 = 0$$. This explains the folded shape at the bottom-right corner of the graph.