Definite Integration Question 16

Question 16

  1. 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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Answer:

Correct Answer: 16. (b)

Solution:

  1. Here, $f(x)=(1-x)^{2} \cdot \sin ^{2} x+x^{2} \geq 0, \forall x$.

and $g(x)=\int_{1}^{x} \frac{2(t-1)}{t+1}-\log t \quad f(t) d t$

$\Rightarrow \quad g^{\prime}(x)=\frac{2(x-1)}{(x+1)}-\log x \cdot \underbrace{f(x)}_{+\mathrm{ve}}$

For $g^{\prime}(x)$ to be increasing or decreasing,

let $\varphi(x)=\frac{2(x-1)}{(x+1)}-\log x$

$\varphi^{\prime}(x)=\frac{4}{(x+1)^{2}}-\frac{1}{x}=\frac{-(x-1)^{2}}{x(x+1)^{2}}$

$\varphi^{\prime}(x)<0$, for $x>1 \Rightarrow \varphi(x)<\varphi(1) \Rightarrow \varphi(x)<0$

From Eqs. (i) and (ii), we get

$$ g^{\prime}(x)<0 \text { for } x \in(1, \infty) $$

$\therefore g(x)$ is decreasing for $x \in(1, \infty)$.



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