Is f(x,y) = (x^2) * exp(y) a convex or concave function?

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Discussion Overview

The discussion centers on the convexity or concavity of the function f(x,y) = (x^2) * exp(y). Participants explore the implications of the Hessian matrix and graphical representations, examining the definitions of convex and concave functions.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant calculates the Hessian of the function and concludes it is negative, suggesting that f(x,y) is concave, but notes a discrepancy with graphical plotting indicating convexity.
  • Another participant disagrees with the initial conclusion, asserting that the graph appears concave and emphasizes the definitions of convex and concave functions.
  • A third participant provides a definition of convexity, stating that a function is convex if the line segment connecting two points on the graph lies above the graph.
  • A later reply warns about the interpretation of the Hessian in the context of multivariable functions, suggesting the presence of a saddle point instead.

Areas of Agreement / Disagreement

Participants express differing views on the convexity or concavity of the function, with no consensus reached regarding the correct classification.

Contextual Notes

Participants highlight the complexity of analyzing multivariable functions and the potential for misinterpretation of the Hessian, as well as the importance of understanding the definitions of convexity and concavity.

mohitp
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Could anyone comment on the convexity of

f(x,y) = (x^2) * exp(y) ... i.e. x square into e to the power y.

I did try to find Hessian of the same and the value I get is :

Hessian(x,y) = -2 * x^2 * exp(2y)... which looks <= 0 for all x and y.

I assume this should imply f(x,y) is concave. However when I plot this function using a 3D graph plotter it seems convex .

try
hp://ww.livephysics.com/ptools/online-3d-function-grapher.php

for plotting function.

I am sure I am making a simple mistake or something.

Any help would be useful.
 
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I don't know why you would say that, even a rough graph looks concave to me. Using the website you give (you are missing a few letters) it looks concave. Are you clear on the difference between "convex" and "concave"? If, for any two points on the graph, the line segment between this is above the graph, it is "concave". It the line segment between two points on the graph is below the graph, then it is "convex".

(Some texts say "concave upward" and "concave downward" rather than "concave" and "convex".)
 
Hi,

Isn't f(x) convex if :

f( a*x1 + (1-a) * x2 ) <= a * f(x1) + (1-a) * f(x2)


this would imply graph must be below line connecting two points for function to be convex.
 
Be careful about Hessian, that is a multivariable function ! and if you work it out that is exactly what it is called a saddle point.
 

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