A superadditive function f(-) satisfies the following property f(x_1 + x_2) >= f(x_1) + f(x_2) Which of the following functions is a superadditive function for x >= 1? A. e^x B. sqrt(x) C. 1/x D. e^(-x)

GATE 2020 · General Aptitude · Functions · medium

Answer: e^x (Option A) is the only superadditive function for x >= 1.

1. Test e^x for superadditivity: For x1, x2 >= 1, let a = e^x1 >= e >= 2.718 and b = e^x2 >= e. We need a*b >= a + b, i.e., a*b - a - b >= 0, i.e., (a-1)(b-1) >= 1. Since a >= e > 1 and b >= e > 1, (a-1)(b-1) >= (e-1)^2 > 1. The inequality holds.
2. Verify sqrt(x) fails: By the triangle inequality / concavity of sqrt, sqrt(a+b) <= sqrt(a) + sqrt(b). So sqrt is subadditive, not superadditive.
3. Verify 1/x and e^(-x) fail: 1/(a+b) < 1/a + 1/b for positive a, b. Also e^(-(a+b)) = e^(-a)*e^(-b) < e^(-a) + e^(-b) for positive a, b (since product of fractions < sum). Both fail.