The partial derivatives are
fx (x, y) = πy cos(πx)
fy (x, y) = sin(πx).
Critical points are solutions of the two equations
πy cos(πx) = 0 ,
sin(πx) = 0 .
The second equation has the infinitely many solutions
x = k , k ∈ Z,
and since cos(kπ) ≠ 0, the first equation implies that y = 0. Thus critical points are (k, 0) with k ∈ Z
To determine the type of the critical points we need the second derivatives
And therefore A = 0, C = 0 and
Since AC − B2 = −π2 < 0 all critical points are saddle points.
