Mathematics - Application of Derivative Question with Solution | TestHub

(A) For $x>0$, $x\log x=3-x$ $\iff g\left(x\right):=$ $\log x-\frac{3}{x}+1=0$. $g$ is continuous on $(0,\infty );g(1)=-2<0$ and $g(3)=\log 3>0$, so by the intermediate value theorem $g$ has a root in (1,3), i.e. the equation has a root in (1,3). (B) $f^{\prime }(x)=4a{x}^3$ $+3b{x}^2+2cx+d$ $\Rightarrow f(x)=a{x}^4$ $+b{x}^3+c{x}^2+dx+e$; $\therefore$$f(0)=f(3)$ as $27a+9b+3c+d=0$, so a root lies in (0,3). (C) $\frac{f(b)-f(a)}{b-a}$ $=f^{\prime }\left(\sqrt{3}\right)$ $=\frac{2}{3}$ $\Rightarrow$ $\frac{ab-1}{ab}$ $=\frac{2}{3}$ $\Rightarrow ab=3$: interval (1, 3). (D) $\frac{f(b)-f(a)}{b-a}$ $=f^{\prime }\left(\frac{1}{2}\right)$ $=1$ $\Rightarrow a+b=1$: interval (0, 1).