Show that there are two curves gamma1nbspand gamma2nbspthat

Question

Supppose F(z) is holomorphic near z = z₀ and F(z₀) = F' (z₀) = 0, while F"(z₀) ≠ 0. Show that there are two curves Γ₁ and Γ₂ that pass through z₀, are orthogonal at z₀, and so that F restricted to Γ₁ is real and has a minimum at z₀, while F restricted to Γ₂ is also real but has a maximum at z₀.

[Hint: Write F(z)=(g(z))² for z near z₀, and consider the mapping z ? g(z) and its inverse.]