That is, we must have and . Dividing the second equation by the first, so y= 3x. x+ y= x+ 3x= 4x= 3 so x= 3/4 and y= 9/4. (3/4, 9/4) is the only critical point for f(x,y) on this line.
In order to be sure that a given function has both maximum and minimum values on a set, that set must be a "closed, bounded" set but that is not what is happening here.Therefore, no gradient vector of f will have the same direction as the gradient vector of x+y=3. Does that mean there is no maximum or minimum? Does this also mean that the constraint has to be a closed function? (Like a circle or something instead of a line)?