Stationary points occur when
dxdy=0.
Maximum points have
dxdy=0 and
dx2d2y<0.
Minimum points have
dxdy=0 and
dx2d2y>0.
Stationary points of inflection have
dxdy=0 and
dx2d2y=0.
Non-stationary points of inflection have
dxdy=0 and
dx2d2y=0.
Decreasing sections of curves have
dxdy<0 and increasing sections of curves have
dxdy>0.
Concave sections of curves have
dx2d2y<0 and convex sections of curves have
dx2d2y>0.