Find extrema of $f_a(x)=\vert x-a\vert^2$ on $\overline{B_1(0)}$

Question

Let $\overline{B_1(0)}\subseteq\mathbb R^3$ be the closed unit-sphere and $a\in\mathbb R^3$.

Find all extrema of the function $f_a(x)=\vert x-a\vert^2$ on $\overline{B_1(0)}$ depending on $a$.

I need tips/hints how to approach such a problem.

Answer 1

Draw a picture: you want to minimize/maximize the distance from $x$ to $a$. It is basically a plane problem, due to symmetry: draw the unit cirle and the point $a$. What is the point of the circle closest to $a$? And what is the farthest?

You'll have to distinguish three cases: (1) $a$ is inside the ball; (2) $a$ is on the boundary of the ball; (3) $a$ is outside the ball.