Find maxima and minima for $f(x,y)=x-2y$ with the constraint $g(x,y)= 4x^2+y^2-17$ using Lagrange
I'm stuck on lagrange on this problem. I get $<1,-2>= \lambda<8x,2y>$ And solve that to get $y=8x$ but don't know what to do from here
There is also a part b which wants how much would f change if g increased by 1 and it would be $1/\lambda$ right?
Note that $4x^2+y^2-17$ is not a constraint. I will assume that the constraint is $4x^2+y^2-17=0$. It's not enough to do $(1,-2)=\lambda(8x,2y)$; $(x,y)$ must also be such that $g(x,y)=0$. So, you have three equations:$$\left\{\begin{array}{l}1=8\lambda x\\-2=2\lambda y\\4x^2+y^2=17.\end{array}\right.$$Its solutions are$$x=-\frac12,y=4,\lambda=-\frac14\quad\text{and}\quad x=\frac12,y=-4,\lambda=\frac14.$$