How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

Test of Mathematics Solution Subjective 84 - Comparing Equations

Test of Mathematics at the 10+2 Level

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.


Show that there is exactly one value of \(x\) that satisfies the equation:

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


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.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.