1. **Understand the problem:** We are asked to order the derivative values (slopes) of the function $f$ at points A, B, C, and D from smallest to largest. 2. **Recall that the derivative at a point is the slope of the tangent line to the curve at that point.** 3. **Analyze the graph description:** - Point B is a local maximum, so the slope at B is zero (flat tangent). - Point C is a local minimum, so the slope at C is also zero. - Point A is before B and is on an increasing part, so the slope at A is positive but less than B's maximum slope. - Point D is after C and rises sharply, so slope at D is large positive. 4. **Ordering the slopes:** - At B and C, slopes are zero. - At A, slope is positive but smaller than at D. - At D, slope is the largest positive. Since B and C both have zero slope, we need to check if one slope is smaller than the other: - At B (local max), slope changes from positive to negative, so slope at B is 0. - At C (local min), slope changes from negative to positive, so slope at C is 0. As zero slope values are equal, but the problem suggests an order, based on the curve geometry, slope at C is increasing after minimum, so slightly positive or zero, whereas slope at B is zero but after maximum, slope decreases. So slope at C could be considered slightly larger than at B. Hence the order of slopes from smallest to largest is: $$\text{slope at B} < \text{slope at C} < \text{slope at A} < \text{slope at D}$$ 5. **Match with the options:** - (a) B, C, D, A - (b) A, B, C, D - (c) C, B, A, D - (d) D, C, B, A Only option (a) matches the order except swapping D and A. Since D's slope is largest positive, it comes last. A's slope is smaller positive. Correct order is **B, C, A, D**, but that is not among options. Considering slopes carefully and standard behavior: - Slope at B (max point) is zero. - Slope at C (min point) is zero. - Slope at A is positive (because function is increasing towards B). - Slope at D is largest positive slope. Hence from smallest to largest slopes: $$0(B) = 0(C) < \text{slope at A} < \text{slope at D}$$ Assuming equality, B and C share smallest slopes, if forced to order B then C: Therefore the answer is (a) **B, C, D, A** if we consider their suggested order. But D's slope should be larger than A. Given the description states D has largest positive slope, a likely best answer is (a) with error in D and A positions swapped. **Final answer:** Option (a) B, C, D, A (interpreting given options).