DSE 2016 Q17

[eco'17]

Consider a set {(x,y) R+ X R| Y<= lnx-e^x}. This set is:
(a) a linear subspace of R2=x
(b) convex
(c) convex and a linear subspace of R2
(d) neither convex, nor a linear subspace of r2

while I know that the given set is not a linear subspace of R2 because this will will not contain 0 vector, because lnx and e^x being symmetric along y=x will not intersect each other, but I don't know how to work for convexity of the set.
i know the definition but I don't know how to apply that
please help
thanks in advance!

[shiva]

This y is the sum of two concave functions, lnx is a concave function and -e^x is also a concave function. so this will be a concave function. The option must be d - neither linear subspace nor convex.

[eco'17]

the answer is (b)
that is, it is convex.
also, I think convexity of a function and set are two different things thus we can not apply this logic here.
I'm not sure though.
but the answer is (b) surely because its given on DU's website itself.

[eco'17]

hey!, I got the answer, thanks for leading me!
If any f(x) is concave, then y<f(x) will necessarily be a convex set, you were almost there.
thanks for your help.

[shiva]

Yes, I got confused.

the given relation is a concave function, whereas the set of points satisfying y<= lnx -e^x will be a CONVEX SET. if you plot the graph, you will get a inverse u shaped curve (which is a concave function) under the all the (x, y) will satisfy the concept of convex set.

[Seecha]

How you havr done 20th question of same paper  that infimum related question ?

[eco'17]

even I didn't get the answer, i will reply if I manage to solve it.

[Econ17]

CONTENTS DELETED

The author has deleted this message.

[eco'17]

since e^x and lnx are inverse of each other, they will never intersect, hence the given set will never have a zero vector. Since, it is important for any set to be subspace of r^2 to have zero vector in it. Thus, the given vector is not a subspace.
this is what my understanding says but I'm not sure.

[Econ17]

CONTENTS DELETED

The author has deleted this message.

[eco'17]

I can't get you, does this fact brings any change to the answer?