Differential Equations 1 Question 22
22. Which of the following is true?
(a) $g$ is increasing on $(1, \infty)$
(b) $g$ is decreasing on $(1, \infty)$
(c) $g$ is increasing on $(1,2)$ and decreasing on $(2, \infty)$
(d) $g$ is decreasing on $(1,2)$ and increasing on $(2, \infty)$
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Solution:
- Here, $f(x)=(1-x)^{2} \cdot \sin ^{2} x+x^{2} \geq 0, \forall x$
and $g(x)=\int _1^{x} \frac{2(t-1)}{t+1}-\log t \quad f(t) d t$
$\Rightarrow g^{\prime}(x)=\frac{2(x-1)}{(x+1)}-\log x \cdot \underbrace{f(x)} _{+ \text {ve }}$
For $g^{\prime}(x)$ to be increasing or decreasing.
Let $\varphi(x)=\frac{2(x-1)}{x+1}-\log x$
$$ \varphi^{\prime}(x)=\frac{4}{(x+1)^{2}}-\frac{1}{x}=\frac{-(x-1)}{x(x+1)^{2}} $$
$\varphi^{\prime}(x)<0, \forall x>1$
$$ \begin{array}{ll} \Rightarrow & \varphi(x)<\varphi(1) \\ \Rightarrow & \varphi(x)<0 \end{array} $$
From Eqs. (i) and (ii), $g^{\prime}(x)<0, x \in(1, \infty)$
$\therefore g(x)$ is decreasing on $x \in(1, \infty)$.
