Question

For all points on the curve x^2 - siny=y+4 where y=pi/2, is the curve concave up or concave down? Why?

Answer 1

if y = π/2, then
x^2 - 1 = π/2 + 4
x = ±√(π/2 + 5)
so, since
y' = x sec^2(y/2)
y" = 2x^2 sin(y/2) sec^5(y/2)
just check the sign of y" at the given values of x.