Browse Questions

# Find the intervals in which the following functions are strictly increasing or decreasing: $(c) \;-2x^3-9x^2-12x+1$

Toolbox:
• A function $f(x)$ is said to be a strictly increasing function on $(a,b)$ if $x_1 < x_2\Rightarrow f(x_1) < f(x_2)$ for all $x_1,x_2\in (a,b)$
• If $x_1 < x_2\Rightarrow f(x_1) > f(x_2)$ for all $x_1,x_2\in (a,b)$ then $f(x)$ is said to be strictly decreasing on $(a,b)$
• A function $f(x)$ is said to be increasing on $[a,b]$ if it is increasing (decreasing) on $(a,b)$ and it is increasing (decreasing) at $x=a$ and $x=b$.
• The necessary sufficient condition for a differentiable function defined on $(a,b)$ to be strictly increasing on $(a,b)$ is that $f'(x) > 0$ for all $x\in (a,b)$
• The necessary sufficient condition for a differentiable function defined on $(a,b)$ to be strictly decreasing on $(a,b)$ is that $f'(x) < 0$ for all $x\in (a,b)$
Step 1:
Given :$f(x)=-2x^3-9x^2-12x+1$
Differentiating w.r.t $x$ we get,
$f'(x)=-6x^2-18x-12$
$\qquad=-6(x^2+3x+2)$
On factorizing we get,
$f'(x)=-6(x+1)(x+2)$
When $f'(x)=0\Rightarrow -6(x+1)(x+2)=0$
Hence $x=-1$ or $x=-2$
Step 2:
The point $x=-2$ and $x=-1$ divide the real number line into three intervals.
(i.e) $(-\infty,-2),(-2,-1),(-1,\infty)$
Consider the interval $(-\infty,-2)$ (i.e) $\infty < x < -2$
Then $(x+1)$ and $(x+2)$ are negative.
The product of the two factors is positive.
$\Rightarrow f'(x)$=(-)(-)(-)=negative.
$\Rightarrow f(x)$ is decreasing in $(-\infty,-2)$
Step 3:
Next consider the interval $(-2,-1)$
(i.e)$-2 < x < -1$
Here $(x+1)$ is negative and $(x+2)$ is positive.
Therefore $f'(x)$ the factor of these two factors is negative.
$\Rightarrow f'(x)$=(-)(+)(+)=negative.
Hence $f(x)$ is decreasing in $(-2,-1)$
Step 4:
Next consider the interval $(-1,\infty)$
(i.e)$-1 < x < \infty$
The factors $(x+1)$ and $(x+2)$ are both positive.
Hence their product is also positive.
$\Rightarrow f'(x)$=(-)(+)(+)=negative.
Hence $f(x)$ is decreasing in $(-1,\infty)$
Hence $f(x)$ is increasing for $-2 < x < -1$ and strictly decreasing for $x < -2$ and $x > -1$