Select the graph that schematically represents BOTH y = x^m and y = x^(1/m) properly in the interval 0 <= x <= 1, for integer values of m, where m > 1. A. [Graph A: y = x^(1/m) is the upper curve (bows upward) and y = x^m is the lower curve (bows downward), both passing through (0,0) and (1,1)] B. [Graph B: y = x^m is the upper curve and y = x^(1/m) is the lower curve] C. [Graph C: y = x^m is the upper curve and y = x^(1/m) is the lower curve, curves close together] D. [Graph D: y = x^(1/m) is the upper curve but both curves appear concave toward the origin]
GATE 2020 · General Aptitude · Curves · medium
Answer: Option A is correct: x^(1/m) is the upper curve and x^m is the lower curve in [0, 1] for m > 1.
- Test a specific value to compare x^m and x^(1/m): x^m = (0.5)^2 = 0.25 and x^(1/m) = (0.5)^(1/2) = sqrt(0.5) approx 0.707. Since 0.25 < 0.707, we have x^m < x^(1/m) for x = 0.5.
- Verify the general inequality for all x in (0,1): Since m > 1, the exponent m makes the value smaller (x^m < x) and the exponent 1/m makes the value larger (x^(1/m) > x). So the order is x^m < x < x^(1/m) for all x in (0,1).
- Match to the correct graph: The correct graph must show: x^(1/m) as the upper curve (bowing toward the y-axis / concave toward origin) and x^m as the lower curve (bowing toward the x-axis). Option A is the only graph with x^(1/m) above x^m in this correct configuration.