Let f(r) = e^(-1) and A(t) denote the area of the region bounded by f(r) and the X-axis, when r varies from -1 to t. Which of the following statements is/are TRUE?
I. f is not bounded in [-1, 1]
II. A is not bounded in [-1, 1]
III. A is nonzero and finite
A. II only
B. III only
C. II and III only
D. I, II and III
GATE 2015 · Engineering Mathematics · Continuity · medium
Answer: C. II and III only
Identify f and its properties: f(r) = 1/e is a constant. Constants are continuous and bounded everywhere. So Statement I ('f is not bounded in [-1,1]') is FALSE.
Compute the area function A(t): A(t) = (t+1)/e. As t -> infinity, A(t) -> infinity. So A is not bounded on its full domain. Statement II ('A is not bounded') is TRUE.
Verify Statement III: Since 1/e > 0 and t > -1, A(t) = (t+1)/e > 0 (nonzero). For any fixed finite t, A(t) is a finite number. So A is nonzero and finite. Statement III is TRUE.
Select the correct option: I is FALSE, II is TRUE, III is TRUE. The answer is C (II and III only).