Tangent plane

In the n-dim case, the tangent lines to the graph of a function are replaced with tangent planes, or, more precisely, tangent hyperplanes.

They are horizontal at the extreme points:

• the partial derivatives are $0$ as numbers, $\frac{∂f}{∂x_i} (a) = 0$ for $i = 1, ..., n$, or
• the gradient is $0$ as a vector, $grad f(a) = 0$, or
• the derivative is $0$ as a matrix, $\nabla f(a) = 0$.

For a more general approach, see Tangent space.