Find the maximum and minimum values of the function f(x)=x^3-9 x^… - Vidyadip

Question

Find the maximum and minimum values of the function $f(x)=x^3-9 x^2+24 x$

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Answer

Given : $f(x)=x^3-9 x^2+24 x$
Differentiating $\text{w.r.t.} \ x,$
$f^{\prime}(x)=3 x^2-18 x+24$
$=3\left(x^2-6 x+8\right)....(1)$
and $ f^{\prime \prime}(x)=6 x-18 ....(2)$
For extreme values of $f(x)$, let $f^{\prime}(x)=0$
$\therefore 3\left(x^2-6 x+8\right)=0$
$\therefore 3(x-2)(x-4)=0$
$\therefore x=2, x=4$
$\therefore$ The stationary points are $x=2$ and $x=4$
Now $\left.f^{\prime \prime}(x)\right|_{x=2}=12-18=-6>0$
and $\left.f^{\prime \prime}(x)\right|_{x=4}=24-18=6>0$
$\therefore f(x)$ is maximum at $x=2$
and minimum at $x=4$
$\therefore$ Maximum value of
$f(x)=f(2)=8-36+48=20$
$\therefore$ Minimum value of
$f(x)=f(4)=64-144+96=16$

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