A superadditive function f(·) satisfies f(x₁+x₂) ≥ f(x₁)+f(x₂). Which of the following functions is superadditive for x > 1?
Options:
(A) eˣ
(B) √x
(C) 1/x
(D) e⁻ˣ
Official Answer Key:
A
Step-by-Step Solution & Explanation
Test eˣ: e^(x1+x2) = e^x1·e^x2 ≥ e^x1+e^x2 whenever both e^x1, e^x2 > 1 (true for x1,x2>1, since product of two numbers each >1 exceeds their sum only when both exceed roughly 2, but more rigorously eˣ is convex and grows faster than linear, satisfying the superadditive property for x>1; the other options are concave/decreasing and fail the inequality).
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