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  4. If y=3x-1+3-x-1 ( x real), then the least value of y is

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Q. If $ y={{3}^{x-1}}+{{3}^{-x-1}} $ ( $ x $ real), then the least value of y is

JamiaJamia 2006

Solution:

Key Idea: For any two numbers a, b, $ \frac{a+b}{2}\ge \sqrt{ab} $ Given $ y={{3}^{x-1}}+{{3}^{-x-1}} $ $ \Rightarrow $ $ y=\frac{{{3}^{x}}}{3}+\frac{{{3}^{-x}}}{3}=\frac{{{3}^{x}}+{{3}^{-x}}}{3} $ ??(i) Now, $ \frac{{{3}^{x}}+{{3}^{-x}}}{2}\le \sqrt{{{3}^{x}}{{.3}^{-x}}} $ $ \Rightarrow $ $ {{3}^{x}}+{{3}^{-x}}\ge 2 $ $ \Rightarrow $ $ \frac{{{3}^{x}}+{{3}^{-x}}}{3}\ge \frac{2}{3} $ $ \Rightarrow $ $ y\ge \frac{2}{3} $ [from(i)]