Assumptions Math Example 4

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Example 4

medium
A theorem states: 'For all a,b>0a, b > 0, a+b2≥ab\frac{a+b}{2} \ge \sqrt{ab} (AM-GM inequality).' Identify the key assumption and give a counterexample showing the result fails without it.

Solution

  1. 1
    Key assumption: a>0a > 0 and b>0b > 0 (both strictly positive).
  2. 2
    If the assumption is removed and we allow a=−4a = -4, b=−1b = -1: −4+(−1)2=−2.5\frac{-4+(-1)}{2} = -2.5 and (−4)(−1)=4=2\sqrt{(-4)(-1)} = \sqrt{4} = 2. Then −2.5<2-2.5 < 2, violating the inequality.
  3. 3
    So the assumption a,b>0a,b > 0 is essential — the AM-GM inequality does not hold for negative numbers.

Answer

Assumption a,b>0 is essential; counterexample: a=−4,b=−1\text{Assumption } a,b>0 \text{ is essential; counterexample: } a=-4, b=-1
Theorems are conditional: they hold only when their assumptions are satisfied. Testing boundary or negative cases reveals whether assumptions are truly necessary.

About Assumptions

Statements accepted as true without proof that form the starting conditions for a mathematical argument or model.

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