Prove Inequality: (a+2)(b+2) ≥ cd with a^2 + b^2 + c^2 + d^2 = 4

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  • Thread starter anemone
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This is a true statement, so the inequality holds under this condition as well.</p>In summary, the given inequality (a+2)(b+2) ≥ cd can be proved using the Cauchy-Schwarz inequality or the AM-GM inequality. It is significant in mathematics and has applications in various fields. It can also be extended to more than four variables and holds under other conditions such as a^2 + b^2 = c^2 + d^2 = 2.
  • #1
anemone
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Here is this week's POTW:

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Let $a$, $b$, $c$ and $d$ be real numbers with $a^2 + b^2 + c^2 + d^2 = 4$.

Prove that the inequality $(a+2)(b+2) \ge cd$ holds.

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  • #2
No one answered last week's POTW. You can read the solution from other as follows:

$\begin{align*}0 &\leq (2 + a + b)^2 \\&= 4 + 4(a + b) + (a + b)^2 \\&= 8 + 4a + 4b + 2ab + a^2 + b^2 – 4 \\&= 2(2 + a)(2 + b) – c^2 – d^2 \\&\leq 2(2 + a)(2 + b) – 2cd \end{align*}$

$\implies (a+2)(b+2) \ge cd \,\,\text{(Q.E.D.)}$
 

FAQ: Prove Inequality: (a+2)(b+2) ≥ cd with a^2 + b^2 + c^2 + d^2 = 4

1. What is the purpose of proving this inequality?

The purpose of proving this inequality is to show the relationship between the variables a, b, c, and d, and to determine if the given expression is always greater than or equal to cd.

2. How can I prove this inequality?

To prove this inequality, you can use algebraic manipulation and the given information about the variables a, b, c, and d. You can also use mathematical theorems or properties, such as the Cauchy-Schwarz inequality or the AM-GM inequality.

3. What are the possible values of a, b, c, and d that satisfy the given equation?

The possible values of a, b, c, and d that satisfy the given equation are all real numbers that make a^2 + b^2 + c^2 + d^2 = 4 true. This includes positive and negative numbers, as well as fractions and irrational numbers.

4. Is this inequality always true?

No, this inequality is not always true. It depends on the values of a, b, c, and d. If the given equation a^2 + b^2 + c^2 + d^2 = 4 is satisfied, then the inequality (a+2)(b+2) ≥ cd is true. However, if the equation is not satisfied, then the inequality may not hold.

5. What are some real-life applications of this inequality?

This inequality can be applied in various fields such as economics, physics, and engineering. For example, it can be used to determine the minimum production cost of a product given certain constraints, or to analyze the stability of a physical system with multiple variables. It can also be used in optimization problems to find the maximum or minimum value of a function.

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