Solve Cauchy's Inequality: Learn How with Examples

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Cauchy's inequality is often misunderstood due to the form it takes in different contexts. The discussion clarifies that the inequality can involve different arrangements of terms, specifically noting that the direction of the inequality sign can vary. By adding terms to both sides and factoring, the original inequality can be transformed into a different form. Squaring both sides can revert it back to the original inequality, while another approach involves multiplying out terms and applying Cauchy's inequality selectively. Understanding these manipulations is crucial for correctly applying Cauchy's inequality in various mathematical scenarios.
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In the solutions sheet, they had this equation in the attachment as Cauchy's inequality. How can that be? Cauchy's inequality involves a multiplication instead of an addition on the right hand side.
 

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    Cauchy's equation.GIF
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We can't see the equation yet, but check what way the inequality sign is pointing..it doesn't always have to be the same side.
 
I worked it out. What they did was add two bits (to the original inequality) to each side and factorised to get three instead of two parts in the inequality. If you square both sides than you get the original inequality back.
 
Or work the other way: multiply out each term on the left hand side an use Cauchy's inequality on part of it.
 

Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
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