Let f be a function of three (real) variables having continuous partial derivatives. For each direction vector $h = (h_1, h_2, h_3)$ such that $h_1^2+ h_2^2+ h_3^2=1$, let $ D_hf (x, y, z)$ be the directional derivative of $f$ along $h$ at $(x, y, z)$. For a point $(x_0, y_0, z_0)$ where the partial derivative $∂/∂xf (x_0, y_0, z_0)$ is not zero, maximize $D_hf (x_0, y_0, z_0)$ (as a function of h)
Totally stuck. Can somebody help how should I solve the problem?