Lagrange multipliers are used to find max/min of function $f(x,y)$ given a constraint $g(x,y)=c$. They do so by assuming the max/min occurs where there are a tangential intersection.
Consider this example: $f(x,y)$ is two planes intersecting at a line forming a V shaped wedge extending upwards to infinity. $g(x,y) = x^2 + y^2 = 1$ is the unit circle. Clearly the min occurs at the line where the two planes intersect. However, this line is not tangential to the unit circle, it goes right down the middle of it.
According to this example, the assumption that both the max/min occur at a tangent is wrong. Can someone explain?