Lagrange multiplier with line constraint

Question

So I am attempting to minimize the 2-dimensional plane equation

$$p(x,y)=ax+by$$

with the constraint

$$g(x,y)=x+y=c.$$

By the method of Lagrange multipliers I get

$$\nabla p=\lambda\nabla g\iff\begin{bmatrix}a \\ b\end{bmatrix}=\begin{bmatrix}\lambda \\ \lambda\end{bmatrix},$$ so only the case $a=b$ would be possible, which I am not really interested in.

What am I missing here? Thanks in advance!

Answer 1

Think about the situation geometrically: you’re trying to find the lowest point on the intersection of the vertical plane $x+y=c$ with some other non-vertical plane. Their intersection is a line, so there is no lowest point unless the line is horizontal, in which case the entire line is at the same altitude, but that can only happen if the second plane is also horizontal.

Answer 2

Hint:

try to minimize $ax+b(c-x)$. What can you conclude in the case $a\ne b $?