Solve Lagrange Multipliers Problem with x-4y=1 Constraint

[Yankel]

Hello all

I have this problem:

Use Lagrange Multipliers to find the min and max of:

\[f(x,y)=xy^{2}\]

under the constraint:

\[x-4y=1\]

\[-1\leqslant x\leq 2\]

My problem is: I know how to solve if

\[-1\leqslant x\leq 2\]

wasn't given. I calculate the Lagrangian function, find it's derivatives by x,y and lambda, and solve the 3 equations to find all suspicious points, I calculate the function value of them all and see which is smallest and which is largest.

I don't know what I should do with the second constraint:

\[-1\leqslant x\leq 2\]

should I simply verify that each point satisfy this condition, or should I check some boundary points ?

 

[MarkFL]

What I would do here is identify the objective function:

\(\displaystyle f(x,y)=xy^2\)

and the constraint:

\(\displaystyle g(x,y)=x-4y-1=0\)

And using Lagrange we ultimately find the critical points:

\(\displaystyle (x,y)=(1,0),\,\left(\frac{1}{3},-\frac{1}{6}\right)\)

Both points have $x$-values within the second constraint. Now, using the $x$-boundaries of the second constraint, we obtain two more critical points:

\(\displaystyle (x,y)=\left(-1,-\frac{1}{2}\right),\,\left(2,\frac{1}{4}\right)\)

So, you have 4 critical points at which to evaluate the objective function, and the smallest and largest values are your minimum and maximum respectively. What do you find?

Spoiler

Doing this, I get values that agree with W|A (if you account for approximations being done by W|A):

optimize x*y^2 subject to x-4y=1,-1<=x<=2