Match the following

Question

Match the following

Professor PC Thomas Classes · Answer

(A) $f(x)=sin~x+cos~x+2x$
$f^{\prime}(x)=cos~x-sin~x+2>0$ strictly increases on $(-\infty,\infty)$
(B) $f(x)=\frac{x^{2}+x+1}{x^{2}-x+1} \Rightarrow f^{\prime}(x)=\frac{(2x+1)(x^{2}-x+1)-(x^{2}+x+1)(2x-1)}{(x^{2}-x+1)^{2}} = \frac{-2x^{2}+2}{(x^{2}-x+1)^{2}}>0 \Rightarrow x\in(-1,1)$
(C) $f(x)=\frac{x^{2}+x+1}{x^{2}-x+1}$ strictly decreases on $(-\infty,-1)\cup(1,\infty)$
(D) $f(x)=\frac{x^{3}}{x^{4}+27} \Rightarrow f^{\prime}(x)=\frac{(x^{4}+27)(3x^{2})-x^{3}(4x^{3})}{(x^{4}+27)^{2}} =\frac{81x^{2}-x^{6}}{(x^{4}+27)^{2}}\Rightarrow x^{2}(81-x^{4})>0 \Rightarrow x^{4}<81 \Rightarrow x\in(-3,3)$