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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Application of Derivatives

Olympiad workbook · IOQM · Application of Derivatives

Let x, y be real numbers such that x y=1 . Let T and t be the...

Let x,yx, y be real numbers such that xy=1x y=1. Let TT and tt be the largest and the smallest values of the expression (x+y)2−(x−y)−2(x+y)2+(x−y)−2\frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2} If T+tT+t can be expressed in the form mn\frac{m}{n} where m,nm, n are nonzero integers with GCD⁡(m,n)=1\operatorname{GCD}(m, n)=1, find the value of m+nm+n.Let x, y be real numbers such that x y=1 . Let T and t be the largest and the smallest values of the expression

Correct answer

25

Step-by-step solution

xy=1⇒(x+y)2=(x−y)2+4(x+y)2−(x−y)−2(x+y)2+(x−y)−2=(x−y)2−(x−y)+2(x−y)2+(x−y)+2=(x−1x)2−(x−1x)+2(x−1x)2+(x−1x)+2x−1x∈R Let f(t)=t2−t+2t2+t+2,t∈R\begin{aligned} & x y=1 \Rightarrow(x+y)^2=(x-y)^2+4 \\ & \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2}=\frac{(x-y)^2-(x-y)+2}{(x-y)^2+(x-y)+2} \\ & =\frac{\left(x-\frac{1}{x}\right)^2-\left(x-\frac{1}{x}\right)+2}{\left(x-\frac{1}{x}\right)^2+\left(x-\frac{1}{x}\right)+2} \\ & x-\frac{1}{x} \in R \\ & \text { Let } f(t)=\frac{t^2-t+2}{t^2+t+2}, t \in R\end{aligned} Let x, y be real numbers such that x y=1 . Let T and t be the largest and the smallest values of the expression
f(t)max⁡+f(t)min⁡=4+24−2+4−24+2=187=mnm+n=25\begin{aligned} & f(t)_{\max }+f(t)_{\min }=\frac{4+\sqrt{2}}{4-\sqrt{2}}+\frac{4-\sqrt{2}}{4+\sqrt{2}}=\frac{18}{7}=\frac{m}{n} \\ & m+n=25\end{aligned}
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