Question

Determine the intervals in which the function f (x)=$x^4-{8x}^3+{22x}^2-24x+21$ is strictly increasing or strictly decreasing.

Answer 1

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

$f\left(x\right)=4x^3-24x^2+44x-24$

$=4(x-1)(x^2-5x+6)$

$=4(x-1)(x-2)(x-3)$

1 for increasing

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

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

$-+-+$

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$xϵ(1,2)\bigcup ,\infty )$

(ii) for decresing

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

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

$-+-+$

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$xϵ(\infty ,1)\bigcup (2,3)$

$(x-1{)}^5(x-2{)}^3(x-3)>0$

$(\because$ power of $3z$ is $1$ )

$(\because$ Power of $2$ in $3$ ) $