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Mathematical Induction, Inequalities and Recurrences

Quick questions on Inequalities explained: H2 Further Mathematics

5short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is proving an inequality?
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To prove A≥BA \geq B, examine the difference A−BA - B and show it is non-negative, typically by writing it as a square or a sum of squares. For example a2+b2≥2aba^2 + b^2 \geq 2ab follows from a2+b2−2ab=(a−b)2≥0a^2 + b^2 - 2ab = (a-b)^2 \geq 0.
What is the discriminant condition?
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For a quadratic ax2+bx+cax^2 + bx + c with real coefficients, the discriminant Δ=b2−4ac\Delta = b^2 - 4ac controls the roots:
What is q1?
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Prove that x2−6x+10>0x^2 - 6x + 10 > 0 for all real xx. [2 marks]
What is q2?
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Solve x2−x−6≤0x^2 - x - 6 \leq 0. [2 marks]
What is q3?
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State the AM-GM inequality for two non-negative numbers and the equality condition. [1 mark]

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