We now relate the level curves to directional derivatives of $f$.
Because the level curves correspond to equally spaced values of $f$, the directional derivative is largest where where the level curves are closest together.
Thus, the maximal magnitude $|\nabla f|$ is located as shown:
(b) The magnitude of $\nabla f$ is small at points where $f$ isn't very steep in any direction.
The smallest possible value $|\nabla f|$ can take is 0. It occurs if the function is locally flat.
The function $f$ must be flat (at its maximal or minimal value) somewhere inside the innermost level set. Hence $|\nabla f|$ is smallest at a point such as:
(c) We need to relate the partial derivative of $f$ to the level sets of $f$.
