Applications of Partial Derivatives

Critical points

A point $(a, b)$ is called a critical point of $f : R^{2} \to R$ if $\nabla f (a, b) = \tilde{0}$ (or if at least one of $f_{x} (a, b) / f_{y} (a, b)$ does not exist, but $f$ does)

DNE = vertical tangent
0 = horizontal tangent

Unlike single variable functions, we need to solve a system of equations when working with multi-variables

\nabla f = \tilde{0} = {\begin{cases} f_{x} (x, y) = 0 \\ f_{y} (x, y) = 0 \end{cases}

Critical Points of multi-variable functions are an extension of the theory with single variable functions

A critical point of a multivariable function can be:

global maximum ()
global minimum
neither

Fermat's Theorems

If $f (x, y)$ has continuous first partial derivatives and $(a, b)$ is a local maximum or minimum of $f (x, y)$ then $f_{x} (a, b) = 0$ and $f_{y} (a, b) = 0$

H f (x, y) = (\begin{matrix} f_{x x} & f_{x y} \\ f_{y x} & f_{y y} \end{matrix})

Suppose that $f (x, y)$ has continuous second partial derivatives near $(a, b)$ and $f_{x} (a, b) = f_{y} (a, b) = 0$ . If $H f (a, b)$ is

positive definite then $(a, b)$ is a local minimum of $f$
negative definite then $(a, b)$ is a local maximum of $f$

With Eigenvalues:
1.

Quadratic forms:
A function $Q (x, y)$ can be expressed as

Q (x, y) = (x - a, y - b) (\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}) (\begin{matrix} x - a \\ y - b \end{matrix})

A quadratic form is:

positive definite if $Q > 0 o$