anemone said:Find, with proof, the minimum value of $(a+b)(b+c)$ where $a,\,b$ and $c$ are positive real numbers satisfying the condition $abc(a+b+c)=1$.
greg1313 said:\(\displaystyle (a+b)(b+c)=ab+ac+bc+b^2=ac+b(a+b+c)=ac+\dfrac{1}{ac}\)
AM-GM:
\(\displaystyle \dfrac{ac+\dfrac{1}{ac}}{2}\ge\sqrt{ac\cdot\dfrac{1}{ac}}=1\)
\(\displaystyle \implies\min\left[(a+b)(b+c)\right]=2\)
greg1313 said:\(\displaystyle (a+b)(b+c)=ab+ac+bc+b^2=ac+b(a+b+c)=ac+\dfrac{1}{ac}\)
AM-GM:
\(\displaystyle \dfrac{ac+\dfrac{1}{ac}}{2}\ge\sqrt{ac\cdot\dfrac{1}{ac}}=1\)
\(\displaystyle \implies\min\left[(a+b)(b+c)\right]=2\)
kaliprasad said:It has been shown that $(a+b)(b+c) >=2$ but I think it needs to be shown that $(a+b)(b+c) = 2 $ is met for at least one set of a,b,c
I am not telling that you are wrong. what I am telling is that I do not get it
greg1313 said:[sp]I'm not clear on what it is you don't understand. However, try $a=1,b=-1+\sqrt2,c=1$.[/sp]
The purpose of finding the minimum of $(a+b)(b+c)$ is to determine the smallest possible value that the expression can have. This can be useful in various mathematical and scientific applications, such as optimization problems.
The minimum of $(a+b)(b+c)$ can be found by using the AM-GM inequality, which states that the arithmetic mean of a set of numbers is always greater than or equal to their geometric mean. By applying this inequality to the expression, we can find the minimum value.
Yes, a proof for finding the minimum of $(a+b)(b+c)$ using the AM-GM inequality is provided in the question. By assuming that $a,b,$ and $c$ are positive real numbers, we can use the AM-GM inequality to show that the minimum value of $(a+b)(b+c)$ is $ab+bc$. This is achieved by setting $x=a+b$ and $y=b+c$ and applying the inequality $x+y \geq 2\sqrt{xy}$.
Yes, there are other methods for finding the minimum of $(a+b)(b+c)$. One alternative method is to use calculus and find the critical points of the expression. Another method is to use algebraic manipulation and completing the square to rewrite the expression in a form where the minimum value is easily identifiable.
Yes, the AM-GM inequality can be applied to find the minimum of other expressions as well. It is a powerful tool in mathematical optimization problems and can be used to find the minimum of various types of expressions involving real numbers.