Contradiction proof of inequality

8. Use proof by contradiction to prove that, for all positive real numbers \(x\) and \(y\), $$\frac { 9 x } { y } + \frac { y } { x } \geqslant 6$$

1. (a) Prove by contradiction that for all positive numbers \(k\)

$$k + \frac { 9 } { k } \geqslant 6$$ (b) Show that the result in part (a) is not true for all real numbers.

6. Prove by contradiction that, if \(a , b\) are positive real numbers, then \(a + b \geqslant 2 \sqrt { a b }\) \includegraphics[max width=\textwidth, alt={}, center]{4de08317-5fb9-4789-8d57-ccf463224c78-20_2655_1943_114_118}

1. (a) Prove that for all positive values of \(a\) and \(b\)

$$\frac { 4 a } { b } + \frac { b } { a } \geqslant 4$$ (b) Prove, by counter example, that this is not true for all values of \(a\) and \(b\).

11 Line 8 states that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\) for \(a\), \(b \geqslant 0\). Explain why the result cannot be extended to apply in each of the following cases.

1. One of the numbers \(a\) and \(b\) is positive and the other is negative.
2. Both numbers \(a\) and \(b\) are negative.

10

1. 
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2. 
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3. Given that \(a\) and \(b\) are distinct positive numbers, use proof by contradiction to prove that $$\frac { a } { b } + \frac { b } { a } > 2$$ \section*{END OF SECTION A
   TURN OVER FOR SECTION B}

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

The curve \(C _ { 1 }\) has equation $$y = x ^ { 4 } + 10 x ^ { 2 } + 8 \quad x \in \mathbb { R }$$ The curve \(C _ { 2 }\) has equation $$y = 2 x ^ { 2 } - 7 \quad x \in \mathbb { R }$$ Use algebra to prove by contradiction that \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.

Prove by contradiction that, for every real number \(x\) such that \(0 \leqslant x \leqslant \frac{\pi}{2}\), $$\sin x + \cos x \geqslant 1.$$ [4]

Prove by contradiction the following proposition: When \(x\) is real and positive, \(x + \frac{81}{x} \geq 18\). [4]

Prove by contradiction the following proposition. When \(x\) is real and positive, $$4x + \frac{9}{x} \geq 12.$$ The first line of the proof is given below. Assume that there is a positive and a real value of \(x\) such that $$4x + \frac{9}{x} < 12.$$ [3]