Finding the Minimum Value of a Complex Expression Using AM-GM Inequality

In this video, we explore how to find the minimum value of the expression. x² + 1/xy + 1/(x²-xy), where x,y positive. We use the powerful Arithmetic Mean-Geometric Mean inequality to break down the problem step-by-step. Follow along as we simplify the expression, apply the inequality, and solve for the conditions that give us the minimum value. Whether you're preparing for a math competition or just want to sharpen your problem-solving skills, this video will walk you through the process in a clear and detailed way. Don't forget to like, share, and subscribe for more math problem-solving content!

[ronbannon]

I like the AM-GM method, but you can also do this by rewriting it as a quadratic, leading to 4 as the minimum.

[eitanhaim]

Hallo Great teacher. Can you please check my solution. I think that I got a smaller minimum.
1/xy + 1/x(x-y)= 1/x[1/y +1/(x+y)]=1/(x-y)y.
Now adding them together by Am Gm gives a smaller solution than 4.