Question

Let f and g are two real valued differentiable functions satisfying.
$f\left(x\right)=\underset{Lt}{\alpha \to 0}\frac{1}{{\alpha }^4}{\int }_0^{\alpha }\frac{(e^{x+t}-e^x)\left(l{n}^2\right(t+1\left)\right)}{2t^2+3}dtand{\int }_0^{x}g\left(t\right)dt=3x+{\int }_x^{0}co{s}^{2}tg\left(t\right)dt$
Range of g(x) is

• A. $[\frac{3}{2},\infty )$
• B. $[\frac{3}{2},3]$
• C. $[-\frac{3}{2},\frac{3}{2}]$
• D. $[-3,3]$

Answer 1

The correct option is B

$[\frac{3}{2},3]$

$g\left(x\right)=\frac{3}{1+co{s}^{2}x};f\left(x\right)=\frac{e^x}{12}$
Range of $g\left(x\right)=[\frac{3}{2},3]$
$f\left(ln6\right)=\frac{e^{ln6}}{12}=\frac{6}{12}$