Question

Find the intervals in which the following function is (a) increasing (b) decreasing
$f\left(x\right)=x^4-8x^3+22x^2-24x+21$

Answer 1

Given,

$f\left(x\right)=x^4-8x^3+22x^2-24x+21$

$f^{\prime }\left(x\right)=3x^3-24x^2+44x-24$

$f^{\prime }\left(x\right)=4(x-1)(x^2-5x+6)$

$f^{\prime }\left(x\right)=4(x-1)(x-3)(x-2)$

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

$4(x-1)(x-3)(x-2)>0$

$1<x<2,3<x<\infty$

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

$4(x-1)(x-3)(x-2)<0$

$2<x<3,x<1$

f(x) increasing range $(1,2)\cup (3,8)$

f(x) decreasing range $(2,3)\cup (-\infty ,1)$