This statement is wrong. In fact, it doesn't even make sense. There is NO line "on the surface off". In fact, there is no "surface" defined by You seem to be assuming that they are talking about the paraboloid "z= f(x,y)" but that is not suggested here. This is just a problem in the xy-plane. You are asked to find points on the line g(x, y)= x+ y= 3 such that f(x,y) has maximum or minimum values. There is no reason to interpret f "geometrically".

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 functionTherefore, no gradient vector offwill 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)?hasboth maximum and minimum values on a set, that set must be a "closed, bounded" set but that is not what is happening here.