Use The Cauchy Condensation Test to prove both of the following statements:
According to course description, material covered includes: linear maps, differentiability, partial derivatives, differentiability of functions whose partial derivatives are continuous, chain rule, Jacobian, inverse and implicit function theorems. Uniform convergence of sequences of functions, Arzela-Ascoli theorem. Basics of Fourier series.
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The expression to minimize is $\left(-s_1-t_2-1\right)^2+\left(-s_2-t_2-1\right)^2+\left(s_1+t_2-1\right){}^2+\left(s_2-t_1+t_2-1\right)^2$, so this is best done by symbolic software.
Answer The planes are at distance $2$, but there is an entire line of critical points
$s_2 = -1 + s_1$ , $t_1 = -2$, $t_2 = -s_1$.
Note A version we worked on in class with (miscopied) $\vec r=s_1\begin{bmatrix}1 \\ 0\\ 0 \\ 1\end{bmatrix}+s_2 \begin{bmatrix}0 \\ 1\\ -1 \\ 0\end{bmatrix}$ led to the pair of plains that interseted at a single point $\begin{bmatrix}1 \\ -1\\ 1 \\ 1\end{bmatrix}$.