Mathematics - Application of Derivative Question with Solution | TestHub

Given, $f\left(x\right)=-2x^3+21x^2-60x+41$

On differentiating w.r.t. $x$, we get

$f^{\text{'}}\left(x\right)=-6x^2+42x-60$

$\Rightarrow f^{\text{'}}\left(x\right)=-6\left(x^2-7x+10\right)$

$\Rightarrow f^{\text{'}}\left(x\right)=-6\left(x-2\right)\left(x-5\right)$

$\Rightarrow x<2 \mathrm{or} x>5, f^{\text{'}}\left(x\right)<0$ i.e., $f\left(x\right)$ is decreasing

and $2<x<5, f^{\text{'}}\left(x\right)>0$ i.e., $f\left(x\right)$ is increasing