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

There are local minima and local maxima points.

This problem will be solved using the Lagrange Multipliers technique.

(see https://tutor.hix.ai)

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://tutor.hix.ai)

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

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To minimize and maximize the function ( f(x, y) = \frac{x^2 + 4y}{e^y} ) constrained to ( 0 < x - y < 1 ), you can use the method of Lagrange multipliers. The Lagrangian function is:

[ L(x, y, \lambda) = \frac{x^2 + 4y}{e^y} + \lambda(x - y - 1) ]

Taking partial derivatives with respect to ( x ), ( y ), and ( \lambda ) and setting them equal to zero, you'll get a system of equations to solve for critical points. After finding the critical points, you can evaluate ( f(x, y) ) at these points and determine which one yields the maximum and which one yields the minimum value.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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