Question

Find the points of maxima, minima and the intervals of monotonicity of the following function:
$y=2x^3-6x^2-18x+7$

Answer 1

$y=2x^3-6x^2-18x+7$

We have to find increasing, decreasing interval and maxima, minima

for increasing $f^{\prime }\left(x\right)>0$ and decreasing $f^{\prime }\left(x\right)<0$

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

$=6(x^2-2x-3)$

$=6(x^2-3x+x-3)$

$=6\left(x\right(x-3)+1(x-3\left)\right)$

$=6(x+1)(x-3)$

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

$6(x+1)(x-3)>0$

so

$x\in (-\infty ,-1)\cup (3,\infty )$

Increasing interval

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

$6(x+1)(x-3)<0$

so

$x\in (-1,3)$

decreasing interval.

Now, for maxima and minima:

$f^{\prime }\left(x\right)=0$ so $6(x+1)(x-3)=0$

at points $x=-1$ and $x=3$

Now to check for local maxima and minima, lets perform second derivative test

$\left(\frac{d^{2}y}{d{x}^2}\right)=12x-12$

at $x=-1,\frac{d^{2}y}{d{x}^2}=-24<0$ [local maxima]

at $x=3,\frac{d^{2}y}{d{x}^2}=36-12=24>0$ [local minima]