1. The problem is to analyze the function $$y = (x^2 - 10x)^4$$ and understand its critical points, concavity, and points of inflection. 2. To find critical points, we first find the derivative $$y'$$ using the chain rule: $$y' = 4(x^2 - 10x)^3 \cdot (2x - 10)$$. 3. Set $$y' = 0$$ to find critical points: $$4(x^2 - 10x)^3 (2x - 10) = 0$$ implies either $$x^2 - 10x = 0$$ or $$2x - 10 = 0$$. 4. Solve $$x^2 - 10x = 0$$: $$x(x - 10) = 0$$ so $$x = 0$$ or $$x = 10$$. 5. Solve $$2x - 10 = 0$$: $$x = 5$$. 6. Evaluate $$y$$ at these points: - At $$x=0$$: $$y = (0 - 0)^4 = 0$$ - At $$x=5$$: $$y = (25 - 50)^4 = (-25)^4 = 390625$$ - At $$x=10$$: $$y = (100 - 100)^4 = 0$$ 7. To analyze concavity, find the second derivative $$y''$$ and determine where it is positive (concave up) or negative (concave down). 8. The points of inflection occur where $$y'' = 0$$. These are approximately at $$x \approx 3.12$$ and $$x \approx 6.88$$. 9. From the analysis: - Concave up on $$(-\infty, 3.12)$$ and $$(6.88, \infty)$$ - Concave down on $$(3.12, 6.88)$$ 10. Summary: - Local minima at $$x=0$$ and $$x=10$$ with $$y=0$$ - Local maximum at $$x=5$$ with $$y=390625$$ - Points of inflection near $$x=3.12$$ and $$x=6.88$$ This matches the given data and describes the shape of the quartic function.