# How do you minimize and maximize #f(x,y)=(x^2+4y)/e^(y)# constrained to #0<x-y<1#?

##### 1 Answer

There are local minima and local maxima points.

#### Explanation:

This problem will be solved using the Lagrange Multipliers technique.

(see https://en.wikipedia.org/wiki/Lagrange_multiplier)

This technique applies to analytic maximization/minimization problems with equality restrictions.

We will transform our problem which is with inequality restrictions into an equivalent one, now with equality restrictions. For this purpose we will introduce the so called slack variables

Minimize/Maximize

subjected to

The set of lagrangian stationary points contains the local minima/maxima points.

The lagrangian is stated as

The lagrangian stationary points are the solutions of

or

This nonlinear system of equations can be solved using a technique similar to Newton-Raphson's

(see https://en.wikipedia.org/wiki/Newton%27s_method)

obtaining

The first and the third are qualified by the restriction

The second and fourth are qualified by the restriction

The qualification is done over

So first and second points are local minima

the third and fourth points are local maxima.

Attached the