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Test of Mathematics Solution Subjective 144 - Finding a Function's Upper Bound

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Test of Mathematics at the 10+2 Level

This is a Test of Mathematics Solution Subjective 144 (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.

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Suppose $ f(x)$ is a real valued differentiable function defined on $ [1,\infty)$ with $ f(1)=1$. Suppose moreover $ f(x)$ satisfies

$ f'(x) = \frac {1}{x^2+f^2(x)}$

Show that $ f(x) \leq 1+\frac{\pi}{4}$ for every $ x \geq 1$


As the question doesn't require us to find an exact solution rather just an upper bound, we can easily find it by manipulating the given statement after establishing certain properties of $ f(x)$.

We see that $ f'(x)>0$ for all $ x$ which means $ f(x)$ is an increasing function.

As the domain is $ [1,\infty)$ we can say that $ f(x)\geq  f(1) $ for all $ x$.

$ => f^2(x)\geq f^2(1)$

$ => x^2+f^2(x)\geq x^2+f^2(1)$         (as $ x^2> 0$)

$ => \frac{1}{x^2+f^2(x)}\leq \frac{1}{x^2+f^2(1)}$

$ => f'(x)\leq \frac{1}{x^2+f^2(1)}$

Integrating both sides from 1 to $ x$

$ => \int_{1}^{x}f'(x)\leq \int_{1}^{x}\frac{1}{x^2+f^2(1)}\leq \int_{1}^{\infty}\frac{1}{x^2+f^2(1)}$

As $ f(1)=1$ we have,

$ => \int_{1}^{x}f'(x)\leq \int_{1}^{\infty}\frac{1}{x^2+1}$

$ => f(x) - f(1) \leq tan^{-1}\infty - tan^{-1}1$

$ => f(x) - f(1) \leq \frac{\pi}{2} - \frac{\pi}{4}$

Substituting $ f(x) = 1$

$ => f(x) \leq  1+\frac{\pi}{4}$

Hence Proved.

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