Proofs involving inequalities (HSC SSCE Mathematics Extension 2): Flashcards
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What is an inequality in mathematics?
What is an inequality in mathematics?
A statement comparing two expressions using <, >, ≤, ≥.
What does AM-GM inequality state?
What does AM-GM inequality state?
Arithmetic mean ≥ geometric mean for non-negative numbers.
What is the Cauchy-Schwarz inequality?
What is the Cauchy-Schwarz inequality?
For vectors, (Σu_iv_i)² ≤ (Σu_i²)(Σv_i²).
What is proof by contradiction?
What is proof by contradiction?
Proving true by showing negation leads to a contradiction.
How do inequalities behave when multiplied by a positive constant?
How do inequalities behave when multiplied by a positive constant?
Direction of inequality is maintained.
What happens to inequalities when multiplied by a negative constant?
What happens to inequalities when multiplied by a negative constant?
The direction of the inequality reverses.
What is the result of adding a constant to an inequality?
What is the result of adding a constant to an inequality?
If a < b, then a + c < b + c.
How can calculus help in proving inequalities?
How can calculus help in proving inequalities?
Use derivatives to find minima and confirm inequalities.
What is the result of the AM-GM inequality for a and b?
What is the result of the AM-GM inequality for a and b?
a + b ≥ 2√(ab) for non-negative a and b.
How can you prove x² + 1 < 0 is impossible?
How can you prove x² + 1 < 0 is impossible?
Assume x² + 1 < 0 leads to contradiction x² < -1.