1. **State the problem:** We need to determine the sign of the derivative $\frac{dy}{dx}$ at points A, B, C, D, and E on a curve. 2. **Recall the meaning of $\frac{dy}{dx}$:** - $\frac{dy}{dx} > 0$ means the curve is increasing (going up) at that point. - $\frac{dy}{dx} = 0$ means the curve has a horizontal tangent (flat slope) at that point. - $\frac{dy}{dx} < 0$ means the curve is decreasing (going down) at that point. 3. **Analyze each point based on the description:** - Point A: On the left side, curve descending steeply $\Rightarrow \frac{dy}{dx} < 0$. - Point B: Local maximum $\Rightarrow \frac{dy}{dx} = 0$. - Point C: Just below x-axis near origin, curve is crossing from negative to positive or vice versa, but no max/min mentioned, so slope is negative or positive? Since B is max and D is min, C is between them and below x-axis, likely slope is negative or positive? Without exact graph, but since curve is going down before B and up after D, at C near origin and below x-axis, slope is negative or positive? Usually, if curve crosses x-axis going down, slope is negative, if going up, slope is positive. Since C is below x-axis near origin, and no max/min, assume slope is negative or positive? We can infer from the curve shape: since B is max left of y-axis and D is min right of y-axis, C near origin below x-axis likely slope is negative (descending). - Point D: Local minimum $\Rightarrow \frac{dy}{dx} = 0$. - Point E: Right of y-axis, curve ascending $\Rightarrow \frac{dy}{dx} > 0$. 4. **Final answers:** - A: $\frac{dy}{dx} < 0$ - B: $\frac{dy}{dx} = 0$ - C: $\frac{dy}{dx} < 0$ - D: $\frac{dy}{dx} = 0$ - E: $\frac{dy}{dx} > 0$