Optimization Through Differentiation
Overview
One of the earliest analytic methods to find an maximum or minimum was developed with the advent of calculus.
Calculus provides the toolset necessary to calculate the tangent to a function at any point.
(see
derivative)
For smooth differentiable
functions, the max or min must occur at a point where the derivative is zero.
(a derivative equal to zero is a necssary condition, but is not sufficient.)
{% \frac{df(x)}{dx} = 0 %}
Demo
The graph shows the tangent line to the parabola. The slider moves the point of tangency. The minimum point is where the tangent is
flat, in other words, the derivative is equal to zero.
Multivariable
The case where the function is question is a function of multiple inputs is a simple extension of the single
variable case. For a multivariable function, each partial derivative (with respect to each input)
must equal zero, whcih can be stated succintly as
{% \nabla f(x_1,...,x_n) = \vec{0} %}
