Basically the core of Lagrange's multiplier says that the solution to a constrained optimization occurs when the contour line of the function being maximized/minimized is tangential to the constraint curve.
I am not able to convince myself of above statement.
Consider below diagram wherein 'f' is some function being maximized/minimized and g(x,y) is the constraint. the contour lines of f and g(x,y) are plotted.
Clearly the contour line which corresponds to local maximum of f is not tangential to g i.e. the gradient vector of f and g don't align at the intersections of f=5 and g.
I am not looking for a rigorous proof or something, just want to get a basic understanding of why gradient of f and g should point in same direction at the local maxima/minima of f?