Convexity/concavity and the function, Z = x²+y²

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In summary, the conversation discusses the function Z = x²+y² and the questions of whether it is continuous and if it is (strictly) increasing or decreasing. The speaker also mentions using the typical definition of concave and convex functions to establish the function's properties. The conversation ends with a question about proving concavity or convexity in the function.
  • #1
ross wils
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very urgent!

consider the function, Z = x²+y²

i need to say i) whether this is continuous

ii) (strictly) increasing or decreasing

i) then establish if the function is a concave, strictly concave, convex or strictly convex function using the typical definition

f[aU + (1-a)V] _> af(U) + (1-a)f(V)


any correspondence before 11.45am GMT would be very appreciated!

ross
 
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  • #2
So what have you tried?
 
  • #3
ive been staring at the problem for days, and that hasn't helped so far.

i thought the left hand side of he inequality might be

f[ax1 + (1-a)x2] ² + [ay1+ (1-a)y2] ² >

with 1 variable the other condition is

f(x) is strictly increasing in X iff for any two points in X, say x1, x2 , such that x2>x1 we have that f(x2) > f(x1), and increasing if you replace the greater than sign with the greater than or equal to sign.

but how two use this for a two variable function of the form Z = x²+y² is still a mystery

still don't really no wat to do
 
  • #4
My first thought was that this is homework but that can't be- because the question simply makes no sense!

It makes no sense to talk about a function being "increasing" because there is no "natural order" on R2.
"(x) is strictly increasing in X iff for any two points in X, say x1, x2 , such that x2>x1 we have that f(x2) > f(x1),"
And what do you mean by "x2> x1" in two dimensions.

As for determining whether f(x,y)= x^2+ y^2 is continuous or not, what properties of continuous functions do you know? It's trivial if you can use the fact that the composition of continuous functions is continuous.
 
  • #5
this homework question has certainly confused me - thanks v much for that reply, but then it goes on to say that looking at that condition and this one, "sum of concave function theorem: if f(x) and g(x) are both concave (convex) functions, then f(x) + g(x) is also a concave (convex) function. if f(x) and g(x) are both concave (convex) functions and in addition either one of them or both are strictly concave (strictly convex) then f(x) + g(x) is also strictly concave (strictly convex)"

this follows on from what you just said. But how does this confirm concavity in the function if indeed it is concave?

for that is my main problem, proving concavity or convexity, be it strictly or no
 

1. What is convexity and concavity?

Convexity and concavity are terms used to describe the shape of a mathematical function. A convex function is one that curves upward, resembling a bowl, while a concave function curves downward, resembling a hill.

2. How can you determine the convexity or concavity of a function?

The convexity or concavity of a function can be determined by looking at its second derivative. If the second derivative is positive, the function is convex. If the second derivative is negative, the function is concave.

3. How does the function Z = x²+y² exhibit convexity and concavity?

The function Z = x²+y² is a convex function. This can be seen by taking the second derivative with respect to both x and y, which yields a positive value for both. Additionally, the graph of this function is a paraboloid, which curves upward and exhibits convexity.

4. Are there any real-world applications of convexity and concavity?

Yes, convexity and concavity have many real-world applications in fields such as economics, engineering, and physics. These concepts are often used to optimize solutions, such as finding the minimum or maximum value of a function.

5. Can a function exhibit both convexity and concavity?

No, a function can only exhibit either convexity or concavity at a given point. However, a function can switch from being convex to concave or vice versa at different points on its graph.

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