Mathematics - Differentiation Question with Solution | TestHub

(A) $g(f(x))=x\Rightarrow g^{\prime }(f(x))=\frac{1}{f^{\prime }(x)}$; for $g^{\prime }(1)\Rightarrow f(x)=1$ at point $(0,1),g^{\prime }(f(x))=\frac{1}{2}$. (B) As $x\to 0,\cos 2x<1$ so $[\cos 2x]=0$, hence the limit $=|x(x-1){|}^0=1$. (C) $x_1+x_2+x_3+x_4=4$, $x_1{x}_2{x}_3{x}_4=1\Rightarrow$ A.M of roots = G.M of roots, $\therefore x_1=x_2=x_3=x_4=1$, giving $a=6,b=-4$, so $a+$ $b=2$. (D) $f^{\prime }(x)=3x^2+2ax$ + $b+c{e}^x$ $\ge 0\forall x$ since $f$ is increasing. At $x=0:f^{\prime }(0)=b+c\ge 0$. The bound is attained: with $a=-\frac{c}{2},b=-c$ (any $c>0$ ), $f^{\prime }(x)=3x^2+c\left(e^x-1-x\right)\ge 0$ because $e^x\ge 1+$ $x$, with equality only at $x=0$ - so $f$ is increasing and $b+c=0$. Hence the minimum value of $b+c$ is 0 .