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Test of Mathematics Solution Subjective 84 - Comparing Equations

This is a Test of Mathematics Solution Subjective 84 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

Problem

Show that there is exactly one value of $$x$$ that satisfies the equation:

$$2 cos^2(x^3+x)=2^x+2^{-x}$$

Solution:

We know that $$cos \;x \leq 1$$ for all $$x \in I\!R$$

$$=> cos(x^3 + x)\leq 1$$

$$=> cos^2(x^3 + x)\leq 1$$

$$=> 2cos^2(x^3 + x)\leq 2$$

Now consider $$2^x$$ and $$2^{-x}$$. By AM-GM inequality we have,

$$2^x+2^{-x}\geq 2$$

So $$2 cos^2(x^3+x)=2^x+2^{-x}$$, only when $$2 cos^2(x^3+x)=2=2^x+2^{-x}$$.

That means $$2^x+2^{-x} = 2, => x= 0$$. So $$x=0$$ being the only solution.

and $$x=0$$ also satisfies $$2 cos^2(x^3+x)=2$$

Thus there is exactly one solution.

Hence Proved.

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