graphs and relations - question regarding obj.func help

[soNasty]

hey guys
i came across this question and am unsure as to how to approach it, can someone please explain it to me?
much appreciated

[#J.Procrastinator]

What exam paper is this from?

[soNasty]

kilbaha 2008

[abcdqdxD]

is the answer D?

[#J.Procrastinator]

is the answer D?

I would go with D too

[soNasty]

Yes the answer is d. But how? What's the point of showing z=0?

[abcdqdxD]

Yes the answer is d. But how? What's the point of showing z=0?

No point really. Look at the function. To maximise it you want lots of x and not so much of y.

[TrueTears]

Answer is D.

Why? Because y = 3x-z, thus the contours of the function z(x,y) all have a gradient of 3 and can only move up and down. To maximise, we use the geometric interpretation of Lagrange Multipliers (tangential), hence D.

http://en.wikipedia.org/wiki/Lagrange_multiplier

[The question shows just one contour, when z = 0, doesn't really relate to the question, all you need is the function z(x,y)]

[#J.Procrastinator]

In the solutions I was given, it says:

Question 9 D
Move line Z = 0 or 3x 􀀁 y = 0 keeping
the movement parallel to the line Z = 0.
Moving to the left passes through P last.
Moving to the right passes through S last.
P would give the minimum and S would
give the maximum.

An alternative is to assign reasonable values for each point and sub it into the objective function to see which point would give the maximum value.

[random_person]

No point really. Look at the function. To maximise it you want lots of x and not so much of y.

Actually I would argue that there is a point. The line gives you the scale of the axis, very important or the answer could be E.

[TrueTears]

Actually I would argue that there is a point. The line gives you the scale of the axis, very important or the answer could be E.

No point, all you need is the functional equation of z(x,y). You know that when the contour of z is tangential to the constraint, then you have either a maximum or minimum (according to LM method). So your choices are either P or S. Then when you move to the right, you move through S last, hence S is the max. The answer can never be T or U since the contours of z will never be tangential to T or U (unless the coefficient of x is 0).

[soNasty]

In the solutions I was given, it says:

Question 9 D
Move line Z = 0 or 3x 􀀁 y = 0 keeping
the movement parallel to the line Z = 0.
Moving to the left passes through P last.
Moving to the right passes through S last.
P would give the minimum and S would
give the maximum.

An alternative is to assign reasonable values for each point and sub it into the objective function to see which point would give the maximum value.

okay so basically its telling us to keep a rule on the line of z=0 and move the ruler to the left, and the leftmost point will be the minimum value (keeping the same gradient), and the rightmost point (in this case S) would give the maximum value, is that right?

and thank you truetears!