Question

Show that for all x ≥ 0 and y ≥ 0 and for conjugate exponents p and q, the following inequality holds xy ≤ {[xᵖ]/p +y[ᵠ]/q}

a) True
b) False

Answer 1

Correct answer is b) False. The inequality does not hold for all x ≥ 0 and y ≥ 0 and for conjugate exponents p and q.

Consider the case where x = 1 and y = 1. The inequality becomes:

1 ≤ {1/p + 1/q}

This inequality does not hold for all values of p and q. For example, if p = 1 and q = 1, then the left-hand side of the inequality is 1 and the right-hand side is 2, which is not less than 1.

Therefore, the inequality does not hold for all x ≥ 0 and y ≥ 0 and for conjugate exponents p and q. Thus, the answer is (b) False.