Question

Which of the following sets are convex?

A. {(x, y) : x² + y² ≥ 1}

B. {(x, y) : y² ≥ x}

C. {(x, y) : 3x² + 4y² ≥ 5}

D. {(x, y) : y ≥ 2, y ≤ 4}

Answer 1

Correct answer is D.

Given sets are

• {(x, y) : x² + y² ≥ 1}

• {(x, y) :y² ≥ x}

• {(x, y) : 3x² + 4y² ≥ 5}

• {(x, y) : y ≥ 2, y ≤ 4}

A convex set, is nothing but whose solution set is in the shape of a convex polygon.

If we map these functions on a graph, we can clearly find the set with a convex solution set.

• f : {(x, y) : x² + y² ≥ 1}

From the graph, it is evident that the solution set which is the shaded region is not convex.

• g:{(x, y) : y² ≥ x}

From the graph, it is evident that the solution set which is the blue shaded region is not convex.

• h : {(x, y) : 3x² + 4y² ≥ 5}

From the graph, it is evident that the solution set which is the grey shaded region is not convex.

• p : {(x, y) : y ≥ 2, y ≤ 4}

From the graph, the dark blue shaded region between the two bright lines is a convex set.