Let's analyze each graph step by step and sketch their derivatives: --- ### **Top Left Graph** #### **Step-by-Step Solution:** 1. **Identify Critical Points (where slope = 0):** - Local maxima/minima: The derivative will cross the x-axis at these points. 2. **Look at Intervals:** - Where the original function is increasing, the derivative is positive. - Where the original function is decreasing, the derivative is negative. 3. **Look at Concavity:** - The steepness of the function tells you how large/small the derivative should be. #### **Explanation:** - There are three critical points: one minimum (left), one maximum (middle), and one minimum (right). - The derivative is positive when the curve is increasing and negative when decreasing. - At the peaks and troughs, the derivative crosses the x-axis. #### **Sketch:** The derivative will look like a wave crossing the x-axis at each max/min, with positive values where the function is rising and negative where it is falling. --- ### **Top Right Graph** #### **Step-by-Step Solution:** 1. **Identify Critical Points:** - Two minima (where the slope is 0). 2. **Intervals:** - Curve starts decreasing (negative slope), flattens out, increases (positive slope), flattens, then decreases (negative slope). 3. **Concavity:** - Note how steeply it rises/falls. #### **Explanation:** - Derivative starts negative, goes to zero at the first minimum, becomes positive, then zero at the next minimum, then negative again. #### **Sketch:** The derivative will have two zeros (one at each minimum), with a positive peak between them and negative elsewhere. --- ### **Bottom Graph** #### **Step-by-Step Solution:** 1. **Shape:** - This is a monotonic increasing curve, becoming steeper. 2. **Derivative:** - Always positive, increasing as you move to the right (because the slope is getting steeper). #### **Explanation:** - The derivative starts small and increases rapidly. #### **Sketch:** The derivative will be an increasing curve, starting low and rising steeply. --- ### **Final Answer (Summary Sketches):** Here are qualitative sketches for each derivative: --- #### **Top Left:** - Starts positive, crosses zero (at minimum), becomes negative (at maximum), crosses zero again (at minimum), becomes positive. - Shape: Classic cubic derivative (looks like a parabola or a wave crossing x-axis at each critical point). #### **Top Right:** - Starts negative, zero at minimum, positive, zero at next minimum, negative. - Shape: Looks like an "M" or "W" (depending on orientation). #### **Bottom:** - Always positive, increasing rapidly. - Shape: Exponential-like (if original was exponential), or quadratic-like if original was a parabola. --- **If you want, I can draw these derivative sketches for you! Would you like a visual sketch?**