Given $$\min k_1x + k_2x^{3/2} + k_3x y^{0.5} + k_4y^{3/2} \\ \text {s.t.} x + y=b \\ \text x>0, y>0, k_i > 0 for 1<=i<=4, b>0$$
Clearly $k_1x$, $k_2x^{3/2}$ and $k_4y^{3/2}$ are convex functions. However, I am not sure if $k_3x y^{0.5}$ is convex. This makes this optimization problem non-convex.
Could someone help me (a) ascertain if $k_3x y^{0.5}$ is concave, (b) if so, then are there any tricks/approximations that I can employ here to get closest convex optimization formulation.