2 marks, −0.667MathematicsConcavity, convexity and quasi-concavityQuasi-concavity and quasi-convexityHardabout 2 min

IIT JAM 2021 Question 21: Concavity, convexity and quasi-concavity

Let $f, g: \mathbb{R} \to \mathbb{R}$ be $f(x) = xe^{-x}$ and $g(x) = x|x|$ . Then, on $\mathbb{R}$ ,

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Answer and worked solution

Answer: (C)

$f(x) = xe^{-x}$ rises to a single peak at $x = 1$ and then falls, so it is single-peaked (quasiconcave) and therefore not quasiconvex.

$g(x) = x|x|$ equals $x^2$ for $x \ge 0$ and $-x^2$ for $x < 0$ , and is strictly increasing throughout; a monotonic function is quasiconvex. So $f$ is not quasiconvex while $g$ is, option (C).

The trap: (A) both

f

and

g

are convex

Neither function is convex on all of $\mathbb{R}$ ( $f$ has an inflection and $g$ is concave on the negative axis), so (A) overstates their curvature; the correct contrast is about quasiconvexity.

Plain-text version

Let f, g: ℝ → ℝ be f(x) = xe^(-x) and g(x) = x|x|. Then, on ℝ, (A) both f and g are convex (B) f is convex and g is not convex (C) f is not quasiconvex and g is quasiconvex (D) neither f nor g is quasiconvex

Also written as: Let f, g: R -> R be f(x) = xe^(-x) and g(x) = x|x|. Then, on R,

Concavity, convexity and quasi-concavity has come up 39 times in 6 exams since 2006.

What this question tests

This question is on Quasi-concavity and quasi-convexity, part of Concavity, convexity and quasi-concavity in Mathematics.

The weaker shape conditions defined by upper or lower level sets.

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