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Given $n$ distinct inputs $t_1,\ldots,t_n$ and prescribed outputs $y_1,\ldots,y_n$, polynomial interpolation finds the polynomial $f\in P_{n-1}$ satisfying

Using the standard basis

write

The coefficients form the coordinate vector

If $f$ has derivatives through order $n$ at $a$, its Taylor polynomial of degree at most $n$ about $a$ is

Suppose $f$ has $n+1$ continuous derivatives on an interval containing $a$ and $x$. If

then

where the integral remainder is

For $x\ne a$, there is also a point $c$ between $a$ and $x$ for which the Lagrange remainder is

Suppose $k$ is an integer with $k\ge2$ and $f$ has $k+1$ continuous derivatives on a neighbourhood of $a$. If

then Taylor's theorem gives

Thus, locally:

• if $k$ is even and $f^{(k)}(a)>0$, $a$ is a local minimum;
• if $k$ is even and $f^{(k)}(a)<0$, $a$ is a local maximum;
• if $k$ is odd, the leading term changes sign across $a$, giving a horizontal point of inflection.

If $f$ is differentiable at $\mathbf{a}=(x_0,y_0)$ and $z_0=f(x_0,y_0)$, then the tangent plane to $z=f(x,y)$ at $(x_0,y_0,z_0)$ is

Let $\mathbf{a}=(x_0,y_0)$, $h_x=x-x_0$ and $h_y=y-y_0$. If $f$ has continuous partial derivatives through order three near $\mathbf{a}$, its Taylor polynomial through total degree three is

The degree-two part can be written with the Hessian as

Let $\mathbf{x}_0=(x_0,y_0)$ be a stationary point. Assume that $f$ has continuous second-order partial derivatives on a neighbourhood of $\mathbf{x}_0$ (so $f$ is $C^2$ there). Define

Then:

• if $D>0$ and $f_{xx}(\mathbf{x}_0)>0$, $\mathbf{x}_0$ is a local minimum;
• if $D>0$ and $f_{xx}(\mathbf{x}_0)<0$, $\mathbf{x}_0$ is a local maximum;
• if $D<0$, $\mathbf{x}_0$ is a saddle point;
• if $D=0$, the test is inconclusive.