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Hello, I am an economics student but this question is purely mathematical.

Imagine the function U(x,y)=x^0.5y^0.5 ( sqrt(x) times sqrt(y) ) in space. If we then imagine the same function equal to a constant we get the 2d functions corresponding to the intersections of horizontal planes with the original 2 variable function, right? (if anyone here is taking economics, these would be the indifference curves). So that for every constant we get a function that is convex to the origin.

Now imagine the function V=U^4=(x^2)(y^2) in space. If we do the same as above for this one, then for every constant we have a function that is concave to the origin, correct?

The problem arises now. My professor says that, the derivative of the one variable functions (the convex and concave ones, the indifference curves) is the ratio of the partial derivatives of the original 2 variable function, which is always the same if the second is a monotonous increasing transformation of the first. Evidently, for the first one: (0.5x^-0.5y^0.5)/(0.5x^0.5y^-0.5) = y/x and for the second one 2xy^2/2x^2y = y/x. The problem is, if the slopes of the functions are the same for every point, then how come one is convex and the other concave? Can this be true? By drawing the curves I can see that it can, but only if we draw the curves that way. Why is it always true? I need to understand what the ratio of the partial derivatives of the original function

Thanks!

Imagine the function U(x,y)=x^0.5y^0.5 ( sqrt(x) times sqrt(y) ) in space. If we then imagine the same function equal to a constant we get the 2d functions corresponding to the intersections of horizontal planes with the original 2 variable function, right? (if anyone here is taking economics, these would be the indifference curves). So that for every constant we get a function that is convex to the origin.

Now imagine the function V=U^4=(x^2)(y^2) in space. If we do the same as above for this one, then for every constant we have a function that is concave to the origin, correct?

The problem arises now. My professor says that, the derivative of the one variable functions (the convex and concave ones, the indifference curves) is the ratio of the partial derivatives of the original 2 variable function, which is always the same if the second is a monotonous increasing transformation of the first. Evidently, for the first one: (0.5x^-0.5y^0.5)/(0.5x^0.5y^-0.5) = y/x and for the second one 2xy^2/2x^2y = y/x. The problem is, if the slopes of the functions are the same for every point, then how come one is convex and the other concave? Can this be true? By drawing the curves I can see that it can, but only if we draw the curves that way. Why is it always true? I need to understand what the ratio of the partial derivatives of the original function

**means**geometrically and logically, what does it represent?Thanks!

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