To find the point on the parabola $y = 8 - x^2$ that is closest to the point $(8, 8)$, we can solve this by minimizing the distance between a general point on the parabola and $(8, 8)$.

1. Define the Distance Function: Let a point on the parabola be $(x, y)$, where $y = 8 - x^2$. The distance $D$ between $(x, 8 - x^2)$ and $(8, 8)$ is: $D = \sqrt{(x - 8)^2 + ((8 - x^2) - 8)^2}$

2. Simplify the Distance Function: Substitute $y = 8 - x^2$ in the distance formula: $D = \sqrt{(x - 8)^2 + (-x^2)^2}$ Expanding: $D = \sqrt{(x - 8)^2 + x^4}$

3. Minimize $D^2$ Instead of $D$: To simplify, we minimize $D^2$ (since minimizing $D$ and $D^2$ yield the same $x$ values). $D^2 = (x - 8)^2 + x^4$ Expanding $D^2$: $D^2 = x^4 + (x - 8)^2$

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