How do I work out whether the function is concave or convex?

[bdx]

The function f(x) =2x−3y is it
a.) concave in x
b.) convex in x
c.) neither concave nor convex in x
d.) both concave and convex in x

 

[Deleted member 4993]

The function f(x) =2x−3y is it
a.) concave in x
b.) convex in x
c.) neither concave nor convex in x
d.) both concave and convex in x

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Please look at your original question and confirm whether the problem as posted is correct (along with your work/thoughts)

 

[Dr.Peterson]

The function f(x) =2x−3y is it
a.) concave in x
b.) convex in x
c.) neither concave nor convex in x
d.) both concave and convex in x

The first thing to do is to find the definitions you were given for "convex in x" and "concave in x". What do they say?

Then try applying that (or a subsequent theorem, if necessary) to this function.

 

[bdx]

The first thing to do is to find the definitions you were given for "convex in x" and "concave in x". What do they say?

Then try applying that (or a subsequent theorem, if necessary) to this function.

Hi Dr Peterson,

thanks for your response, I am studying mathematical finance masters and the questions put to me were literally put like this but I will attach the document I am completing the work for, this is in preparation for the entrance exam (I come from an economics background so this is fairly different from what i am used to)

Thanks again for your help

 

[Dr.Peterson]

The definition of convexity given there is:

A function of a single variable is convex if the line joining two points on the graph of the function lies above the graph.​

A function of a single variable is concave if the line joining two points on the graph of the function lies below the graph.​

They go on to define convexity for a function of more than one variable; but that isn't necessary here. It's unfortunate that they don't explicitly define "convex in x", but I would take that to mean "convex when considered as a function of x alone", which means "think of y as a constant".

Can you now apply the definition to your function? (It's easiest just to sketch a graph, thinking of f(x) =2x−3y as if it were y = 2x - 3k, where k could be any number.)

 

[Dr.Peterson]

I should explicitly tell you that I am waiting for your response, because there is more to say. I expect you to ask about a subtlety in the definition, and perhaps even to discover an error in it (if you look carefully at the formal definition they give after the informal single-variable definition).

To make this clearer, take a look at Wikipedia's informal definition:

In mathematics, a real-valued function defined on an n-dimensional interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph.​

 

[bdx]

Apologies for the delayed response, I should take it to mean that x is strictly convex if -3y is treated like a constant but I am not 100% sure, the example given is perplexing and the rationale of the document isnt easy to extract and apply as I would like.

 

[bdx]

Apologies for the delayed response, I should take it to mean that x is strictly convex if -3y is treated like a constant but I am not 100% sure, the example given is perplexing and the rationale of the document isn't easy to extract and apply as I would like.

P.S thank you for your continued help Dr. Peterson, it is greatly appreciated

 

[Dr.Peterson]

Apologies for the delayed response, I should take it to mean that x is strictly convex if -3y is treated like a constant but I am not 100% sure, the example given is perplexing and the rationale of the document isnt easy to extract and apply as I would like.

I'm not sure what you mean by "x is strictly convex", since the question is about the function, not the variable, and I don't think they used the term "strictly" here. I'd like to know more of what you are thinking.

What does the graph look like, if y is held constant? It's a linear function, right? A function is convex if the straight line joining any two points lies above [or on] the graph. But here, the graph itself is a straight line. If I were to use the word "strictly" here, I would mean that the line lies "strictly above", and not "on", the graph. In that sense, then, this function is not strictly convex (which is what their informal, one-variable definition really says), but it is convex (by the formal definition, which has ≤ in it).

Unfortunately, the question depends heavily on this specific point, on which the text is inconsistent. So yes, it is perplexing, which is why it will need further discussion after you have stated your answer clearly.

This course appears to be a rapid review of material that may be new to at least some students; if any tutoring or office hours are available, you should take advantage of it to help you get up to speed.