CIIM 2014 Problem 5

A analityc function $f:\mathbb{C}\to\mathbb{C}$ is call interesting if $f(z)$ is real along the parabola $Re (z) = (Im (z))^2$ .

a) Find an example of a non constant interesting function. b) Show that every interesting function $f$ satisfy that $f'(-3/4) = 0.$