Mathematics - Application of Derivative Question with Solution | TestHub

$f(x)=x^4-4x^3+12x^2+x-1$ $$\begin{array}{rl}{f}^{\prime }(x)& =4x^3-12x^2+24x+1\\ f^{\prime \prime }(x)& =12x^2-24x+24\\ & =12\left(x^2-2x+2\right)\\ & =12\left\{(x-1{)}^2+1\right\}>0,\text{ for all }x\end{array}$$ $\Rightarrow f^{\prime }(x)$ is increasing. Since, $f^{\prime }(x)$ is cubic and increasing. $\Rightarrow f^{\prime }(x)$ has only one real root and two imaginary roots. $\therefore f(x)$ cannot have all distinct root. $\Rightarrow$ Atmost 2 real roots. Now, $f(-1)=15$, $f(0)=-1$ and $f(1)=9$ $\therefore f(x)$ must have one root in $(-1,0)$ and other in $(0,1)$. $\Rightarrow 2$ real roots.