1. The problem states that the graph of the function $f(x) = 2x^3 - 3x^2 - 12x + 7$ is decreasing at the point $(-1, 2)$. We need to verify this by analyzing the derivative of the function. 2. The derivative $f'(x)$ gives the slope of the tangent line to the graph at any point $x$. If $f'(x) < 0$, the function is decreasing at that point. 3. Compute the derivative: $$f'(x) = \frac{d}{dx}(2x^3) - \frac{d}{dx}(3x^2) - \frac{d}{dx}(12x) + \frac{d}{dx}(7) = 6x^2 - 6x - 12$$ 4. Evaluate the derivative at $x = -1$: $$f'(-1) = 6(-1)^2 - 6(-1) - 12 = 6(1) + 6 - 12 = 6 + 6 - 12 = 0$$ 5. Since $f'(-1) = 0$, the slope of the tangent line at $x = -1$ is zero, indicating a critical point (possible maximum, minimum, or inflection). 6. To determine if the function is decreasing at $x = -1$, check the sign of $f'(x)$ just before and after $x = -1$: - For $x = -1.1$, $$f'(-1.1) = 6(-1.1)^2 - 6(-1.1) - 12 = 6(1.21) + 6.6 - 12 = 7.26 + 6.6 - 12 = 1.86 > 0$$ - For $x = -0.9$, $$f'(-0.9) = 6(0.81) - 6(-0.9) - 12 = 4.86 + 5.4 - 12 = -1.74 < 0$$ 7. The derivative changes from positive to negative at $x = -1$, so the function is increasing before $x = -1$ and decreasing after $x = -1$. Therefore, the graph is decreasing immediately after the point $(-1, 2)$. 8. This confirms the graph is decreasing at the point $(-1, 2)$ as stated. Final answer: The function $f(x) = 2x^3 - 3x^2 - 12x + 7$ is decreasing at $x = -1$ because $f'(x)$ changes from positive to negative there, indicating a local maximum and a decreasing slope after that point.