Question

If $f\left(x\right)=2x^3-9x^2+12x-6$, then in which interval $f\left(x\right)$ is monotonically increasing.

• A. $(1,2)$
• B. $(-\infty ,1)$
• C. $(2,\infty )$
• D. $(-\infty ,1)$ and $(2,\infty )$

Answer 1

The correct option is D $(-\infty ,1)$ and $(2,\infty )$

$f\left(x\right)=2x^3-9x^2+12x-6$

$f^{\prime }\left(x\right)=6x^2-18x+12$

$=6\left(x^2-3x+2\right)$

$=6\left(x^2-2x-x+2\right)$

$=6\left[x\left(x-2\right)-1\left(x-2\right)\right]$

$=6\left(x-2\right)\left(x-1\right)$

$Forcriticalpo\mathrm{int}$

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

$=6\left(x-1\right)x-2=0$

$=x=1,2$

$f^{\prime }\left(x\right)>0\forall x\in \left(-0,1\right)\cup \left(2,\infty \right)$

So, $f\left(x\right)$ monotonic increasing in $\left(-\infty ,1\right)$ and $\left(2,\infty \right)$