Prove Inequality: $\sqrt{ab}+\sqrt{cd}\le \sqrt{(a+d)(b+c)}$

In summary, proving an inequality helps us compare and contrast different values and make informed decisions. The inequality symbol represents a relationship between two values where one is greater than or less than the other. To prove an inequality, we can use the properties of square roots and multiplication. The expression on the left side of the inequality is important as it is used to compare with the expression on the right side. This inequality can be used for all real values of a, b, c, and d, but not necessarily for complex numbers.
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Prove that for positive reals $a,\,b,\,c,\,d$, $\sqrt{ab}+\sqrt{cd}\le \sqrt{(a+d)(b+c)}$.
 
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We have:

$$\sqrt{ab} + \sqrt{cd} \leq \sqrt{(a+d)(b+c)} \\\\
\left ( \sqrt{ab} + \sqrt{cd}\right )^2 \leq \left ( \sqrt{ab + ac + bd + cd} \right )^2
\\\\ab + cd + 2\sqrt{ab}\sqrt{cd}\leq ab + ac + bd + cd
\\\\ac + bd - 2\sqrt{ab}\sqrt{cd}\geq 0
\\\\\left ( \sqrt{ac} \right )^2 + \left ( \sqrt{bd} \right )^2-2\sqrt{ac}\sqrt{bd}\geq 0
\\\\\left ( \sqrt{ac}-\sqrt{bd} \right )^2 \geq 0. $$

Thus the inequality holds.
 

FAQ: Prove Inequality: $\sqrt{ab}+\sqrt{cd}\le \sqrt{(a+d)(b+c)}$

What is the inequality being proven?

The inequality being proven is $\sqrt{ab}+\sqrt{cd}\le \sqrt{(a+d)(b+c)}$.

What is the significance of this inequality?

This inequality is a fundamental concept in mathematics and is used in various fields such as algebra, geometry, and calculus. It allows us to compare the sizes of different numbers and is essential in solving mathematical problems.

How is this inequality proven?

This inequality can be proven using various methods such as algebraic manipulation, geometric proofs, or by using the Cauchy-Schwarz inequality. The specific method used depends on the context and the level of mathematical knowledge of the person proving it.

What are the conditions for this inequality to hold?

In order for this inequality to hold, the numbers a, b, c, and d must be non-negative real numbers. Additionally, the inequality is only valid when the numbers are arranged in a specific order, with a and d being the larger numbers and b and c being the smaller numbers.

Are there any real-life applications of this inequality?

Yes, this inequality has many real-life applications in fields such as economics, physics, and engineering. It is used to optimize resources, analyze data, and make predictions based on mathematical models.

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