1. **State the problem:** We have the function $y = x^2 - k\sqrt{x}$, where $k$ is a constant. We are given the derivative $$\frac{dy}{dx} = 2x - \frac{1}{2}kx^{-\frac{1}{2}}$$ and told that $y$ is decreasing at $x=4$. We need to find the set of possible values of $k$. 2. **Recall the rule:** A function is decreasing at a point if its derivative at that point is less than zero. So, $$\frac{dy}{dx} < 0 \text{ at } x=4.$$ 3. **Substitute $x=4$ into the derivative:** $$\frac{dy}{dx} = 2(4) - \frac{1}{2}k(4)^{-\frac{1}{2}} = 8 - \frac{1}{2}k \cdot \frac{1}{\sqrt{4}} = 8 - \frac{1}{2}k \cdot \frac{1}{2} = 8 - \frac{k}{4}.$$ 4. **Set the inequality for decreasing:** $$8 - \frac{k}{4} < 0.$$ 5. **Solve for $k$:** $$8 < \frac{k}{4}$$ Multiply both sides by 4: $$\cancel{4} \times 8 < \cancel{4} \times \frac{k}{4}$$ $$32 < k$$ 6. **Conclusion:** The function is decreasing at $x=4$ if and only if $$k > 32.$$