To solve this problem, we need to find the point on the parabola $y = 8 - x^2$ that is closest to the point $(8, 8)$.

Solution Outline

1. Define the Distance Function: The distance $D$ between any point $(x, y)$ on the parabola and the point $(8, 8)$ is given by: $D = \sqrt{(x - 8)^2 + (y - 8)^2}$ Since $y = 8 - x^2$, substitute $y$ in terms of $x$ to express $D$ as a function of $x$ alone.

2. Rewrite the Distance Squared: To simplify, minimize $D^2$ instead of $D$, since the minimum of $D^2$ will occur at the same point as the minimum of $D$. Thus, we define: $D^2 = (x - 8)^2 + (8 - x^2 - 8)^2$ Simplifying, this becomes: $D^2 = (x - 8)^2 + (-x^2)^2$ $D^2 = (x - 8)^2 + x^4$

3. Differentiate and Find Critical Points: Differentiate $D^2$ with respect to $x$, set the derivative equal to zero, and solve for $x$ to find the critical points.

4. Evaluate and Determine the Closest Point: Calculate $y$ for the resulting $x$-value(s) and determine which point gives the minimum distance.

I’ll go through these steps now and provide the result.