Question

Find the intervals in which the function $f$ given by $f\left(x\right)=2x^3-3x^2-36x+7$ is

$\left(a\right)$ increasing
$\left(b\right)$ decreasing

Answer 1

$f\left(x\right)=2x^3-3x^2-36x+7$
Differentiating w.r.t $x,$
$f^{\prime }\left(x\right)=6x^2-6x-36$
$\Rightarrow f^{\prime }\left(x\right)=6(x^2-x-6)$
$\Rightarrow f^{\prime }\left(x\right)=6(x-3)(x+2)$
Putting $f^{\prime }\left(x\right)=0$
$6(x+2)(x-3)=0$
$\Rightarrow x=-2,3$

Plotting points on number line

Intervals	Sign of $f^{\prime }\left(x\right)=6(x-3)(x+2)$	Nature of function $f$
$(-\infty ,-2)$	$(-)(-)>0$	increasing
$-2,3$	$(+)(-)<0$	decreasing
$(3,\infty )$	$(+)(+)>0$	increasing

Hence, (a) increasing in
$(-\infty ,-2)\cup (3,\infty )$
(b) decresing in
$(-2,3)$