Let f(x) = 2x + tan^{-1} x and g(x) = log_e(s | vedclass

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(B) Given $f(x) = 2x + 	an^{-1} x$ and $g(x) = \ln(\sqrt{1+x^2} + x)$ for $x \in [0, 3]$.First,find the derivatives:$f'(x) = 2 + \frac{1}{1+x^2}$$g'(

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$\max f(x) > \max g(x)$

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(B) Given $f(x) = 2x + 	an^{-1} x$ and $g(x) = \ln(\sqrt{1+x^2} + x)$ for $x \in [0, 3]$.First,find the derivatives:$f'(x) = 2 + \frac{1}{1+x^2}$$g'(x) = \frac{1}{\sqrt{1+x^2} + x} \cdot \left(\frac{x}{\sqrt{1+x^2}} + 1ight) = \frac{1}{\sqrt{1+x^2} + x} \cdot \frac{x + \sqrt{1+x^2}}{\sqrt{1+x^2}} = \frac{1}{\sqrt{1+x^2}}$.For $x \in [0, 3]$:$f'(x) \in [2 + \frac{1}{1+3^2}, 2 + \frac{1}{1+0^2}] = [2.1, 3]$.$g'(x) \in [\frac{1}{\sqrt{1+3^2}}, \frac{1}{\sqrt{1+0^2}}] = [\frac{1}{\sqrt{10}}, 1] \approx [0.316, 1]$.Since $f'(x) > g'(x)$ for all $x \in [0, 3]$,option $A$ is incorrect.Both $f(x)$ and $g(x)$ are strictly increasing functions on $[0, 3]$.Thus,$\max f(x) = f(3) = 6 + 	an^{-1} 3$ and $\max g(x) = g(3) = \ln(3 + \sqrt{10})$.Since $6 + 	an^{-1} 3 > 6$ and $\ln(3 + \sqrt{10}) \approx \ln(6.16)  g(3)$.Therefore,option $B$ is correct.

List $I$	List $II$
$A. 3x^4 - 2x^3 - 6x^2 + 6x + 1$	$(I)$ has minimum value at $x = 4$
$B. x + \frac{1}{x}, \forall x < 0$	$(II)$ has maximum value at $x = -1$
$C. x^4(7 - x)^3$	$(III)$ has maximum value at $x = 4$
$D. x^4 + (8 - x)^4$	$(IV)$ is decreasing in $[2, \infty)$
	$(V)$ is increasing in $[2, \infty)$

List-$I$	List-$II$
$(P)$ The minimum value of $n$ for which the function $f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]$ is continuous on the interval $[1,2]$,is	$(1)$ $8$
$(Q)$ The minimum value of $n$ for which $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right), x \in R$,is an increasing function on $R$,is	$(2)$ $9$
$(R)$ The smallest natural number $n$ which is greater than $5$,such that $x=3$ is a point of local minima of $h(x)=\left(x^2-9\right)^{n}\left(x^2+2 x+3\right)$,is	$(3)$ $5$
$(S)$ Number of $x_0 \in R$ such that $l(x)=\sum_{k=0}^4\left(\sin \|x-k\|+\cos \left\|x-k+\frac{1}{2}\right\|\right), x \in R$ is not differentiable at $x_0$,is	$(4)$ $6$
	$(5)$ $10$

Let $f(x) = 2x + \tan^{-1} x$ and $g(x) = \log_e(\sqrt{1+x^2} + x)$,$x \in [0, 3]$. Then:

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