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.

Related

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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