Question

Find the intervals in which $f\left(x\right)=2x^3-3x^2-36x+25,x\in R$ is

i. strictly increasing.

ii. strictly decreasing.

Answer 1

$f\left(x\right)=2x^3-3x^2-36x+25$

$⟹f^{\prime }\left(x\right)=6x^2-6x-36=6(x^2-x-6)$

$⟹f^{\prime }\left(x\right)=6(x-3)(x+2)$

For strictly increasing-

$f^{\prime }\left(x\right)>0$

$⟹(x-3)(x+2)>0$

$⟹x<-2$or $x>3$ that is $x\in (-\infty ,-2)\cup (3,\infty )$

For strictly decreasing-

$f^{\prime }\left(x\right)<0$

$⟹(x-3)(x+2)<0$

$⟹-2<x<3$ that is $x\in (-2,3)$