Question
Download Solution PDFWhat is the minimum value of a2x + b2y where xy = c2?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
AM-GM inequality:
If x1. . . Xn ≥ 0, then
\(\Rightarrow \frac{{x1 + x2 + \ldots + xn}}{n}\; \ge \;\sqrt[n]{{x1.x2 \ldots xn}}\)
Calculation:
Let a2x and b2y are two numbers.
Then apply AM-GM inequality
\(AM \ge GM\)
\(\Rightarrow \frac{{{{\rm{a}}^2}{\rm{x\;}} + {\rm{\;}}{{\rm{b}}^2}{\rm{y}}}}{2}\; \ge \;\sqrt[2]{{{{\rm{a}}^2}{\rm{x\;}}{{\rm{b}}^2}{\rm{y}}}}\)
\(\Rightarrow {{\rm{a}}^2}{\rm{x\;}} + {\rm{\;}}{{\rm{b}}^2}{\rm{y\;}} \ge 2\sqrt[2]{{{{\rm{a}}^2}{{\rm{b}}^2}xy}}\)
\(Given\;xy = {c^2}\)
\(\Rightarrow {{\rm{a}}^2}{\rm{x\;}} + {\rm{\;}}{{\rm{b}}^2}{\rm{y}} \ge 2\sqrt[2]{{{{\rm{a}}^2}{{\rm{b}}^2}{c^2}}}\)
\( \Rightarrow {{\rm{a}}^2}{\rm{x\;}} + {\rm{\;}}{{\rm{b}}^2}{\rm{y\;}} \ge {\rm{\;}}2{\rm{abc}}\)
∴ minimum value of a2x + b2y is 2abc
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