# Find the intervals in which the following functions are strictly increasing or decreasing: $(a) \;x^2 + 2x - 5$

$\begin{array}{1 1}(A)\;\text{Strictly decreasing for} x<-1 \text{and strictly increasing for} x>-1. \\ (B)\;\text{Strictly increasing for} x<-1 \text{and strictly decreasing for}x>-1. \\ (C)\;\text{Strictly decreasing for} x<-1 \text{and strictly decreasing for} x>-1. \\ (D)\;\text{Strictly increasing for} x<-1 \text{and strictly decreasing for} x>-1\end{array}$

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 :$f(x)=x^2+2x-5$
Differentiating w.r.t $x$ we get,
$f'(x)=2x+2$
$\qquad=2(x+1)$
The function $f(x)$ will be increasing if $f'(x)>0$
(i.e)$2(x+1)=0$
$\Rightarrow x+1>0$
$x>-1$
Step 2:
The function $f(x)$ will be decreasing if $f'(x)<0$
(i.e)$2(x+1)<0$
$\Rightarrow x<-1$
Hence $f(x)$ is increasing on $(-1,\infty)$ and decreasing on $(-\infty,-1)$
answered Jul 9, 2013
edited Jul 9, 2013