Thread: Convex Set

Show that if f is any convex function and b any real number, then the solution set of the inequality f(x)<=b is a convex set.

A function is convex if.....

f(λX(1) + (1 - λ)X(2)) <= λf(X(1)) + (1 - λ)f(X(2))

where X(1) and X(2) are X subscript 1 and X subscript 2

I have no clue how to do this. Can someone help me out!!!! TIA!!!

Originally Posted by TreeMoney

Show that if f is any convex function and b any real number, then the solution set of the inequality f(x)<=b is a convex set.

A function is convex if.....

f(λX(1) + (1 - λ)X(2)) <= λf(X(1)) + (1 - λ)f(X(2))

where X(1) and X(2) are X subscript 1 and X subscript 2

I have no clue how to do this. Can someone help me out!!!! TIA!!!

Just note that
f(λX(1) + (1 - λ)X(2)) <= λf(X(1)) + (1 - λ)f(X(2))<=λ(b)+(1-λ)b=b
for 0<=λ<=1.