Question

Find the interval in which the following function are strictly increases or decreasing:

$x^2+2x-5$

• A. Increasing for $(-\infty ,-1)$ and decreasing for $(-1,\infty )$
• B. Increasing for $(-\infty ,-2)$ and decreasing for $(-2,\infty )$
• C. Increasing for $(-\infty ,-3)$ and decreasing for $(-3,\infty )$
• D. Increasing for $(-\infty ,-4)$ and decreasing for $(-4,\infty )$

Answer 1

The correct option is A Increasing for $(-\infty ,-1)$ and decreasing for $(-1,\infty )$

$f\left(x\right)=x^2+2x-5$

Step 1 :- Calculating $f^1\left(x\right)$

$f^1\left(x\right)=2x+2(x+1)$

Step 2 :- Putting $f^1\left(x\right)=0$

$2(x+1)=0$

$x+1=0$

$x=-1$

Step 3 :- Pletting point as real line

point $x=-1$ divides the real line into $2$ disjoint interval $(-\infty ,-1)$ & $(-1,\infty )$

Step 4:

Value of x	Intervals	sign of $f^1\left(x\right)=2(x+1)$	Nature of function ${}^{\prime }{f}^{\prime }$
$x<-1$	$x\epsilon (-\infty ,-1)$	$(-)<0$	$f$ is strictly decreasing
$x>-1$	$x\epsilon (-1,\infty )$	$(+)>0$	$f$ is strictly increasing

Hence, $f$ is decreasing for $(-\infty ,-1)$

$f$ is increasing for $(-1,\infty )$