A question on gradients: $f$ must assume equal value at two points

Question

If $\nabla f(x,y,z)$ is always parallel to $x i+y j+z k$, show that $f$ must assume equal values at the points $(0,0,a)$ and $(0,0,-a)$.

Answer 1

We have $$\nabla f(x, y , z) = \lambda(xi + yj + zk)$$

Equating the $z$-components gives $$\frac{\partial{f}}{\partial{z}} = \lambda z \\ \Rightarrow f(x, y, z) = \frac{\lambda z^2}{2} + g(x, y)$$ where $g(x, y)$ is some arbitrary function. By this formula, we obtain $$f(0, 0, a) = \frac{\lambda a^2}{2} + g(0,0) \\ f(0,0,-a) = \frac{\lambda a^2}{2} + g(0,0)$$ which proves the given assertion.