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

[ross wils]

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

 

[StatusX]

So what have you tried?

 

[ross wils]

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

 

[HallsofIvy]

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.

 

[ross wils]

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