Question

A function f(x) satisfies the relation ${\int }_2^{x}f\left(t\right)=\frac{x^2}{2}+{\int }_x^2{t}^{2}f\left(t\right)dt$
Then answer the following
The range of f(x) is

• A. $[0,\infty ]$
• B. $R$
• C. $[-\frac{1}{2},\frac{1}{2}]$
• D. $[\frac{1}{2},\infty ]$

Answer 1

The correct option is C $[-\frac{1}{2},\frac{1}{2}]$

${\int }_2^{x}f\left(t\right)dt=\frac{x^2}{2}+{\int }_x^2{t}^{2}f\left(t\right)dtDiff.Wrt{}^{\prime }{x}^{\prime }f\left(x\right)=x-x^{2}f\left(x\right)\Rightarrow f\left(x\right)=\frac{x}{1+x^2}$
Let $y=\frac{x}{1+x^2}\Rightarrow 1-4y^2\ge 0\Rightarrow yϵ[-\frac{1}{2},\frac{1}{2}]$