Absolute Minimums and Maximums — Question 8

Question 8 – Solution

Strategy. The function is squared distance from $(x,y)$ to $(a,0)$; minimize and maximize the two nonnegative coordinate contributions independently.

Step 1: Minimum The $y^2$ term is minimized uniquely at $y=0$. The closest $x\in[-1,1]$ to $a$ is the clipped value $$x_*= \begin{cases} -1,&a<-1,\\ a,&-1\le a\le 1,\\ 1,&a>1. \end{cases}$$ Hence the unique minimizing point is $(x_*,0)$, and $$\boxed{f_{\min}=\bigl(\max\{|a|-1,0\}\bigr)^2}.$$

Step 2: Maximum The largest possible $y^2$ is $1$, attained at $y=\pm 1$. The endpoint of $[-1,1]$ farthest from $a$ is at distance $|a|+1$. Therefore $$\boxed{f_{\max}=(|a|+1)^2+1}.$$ If $a>0$, the maximizers are $(-1,\pm 1)$; if $a<0$, they are $(1,\pm 1)$.

Step 3: Tie case When $a=0$, both $x$-endpoints are equally far away, so all four corners $(\pm 1,\pm 1)$ maximize $f_0$ with value $2$. The point $(0,0)$ uniquely minimizes it with value $0$. This agrees exactly with the squared-distance interpretation.