Question

The real function $f\left(x\right)=1-x-x^2$ is increasing in $x$ over the interval

• A. $(-\infty ,\frac{-1}{2})$
• B. $(-\infty ,\frac{1}{2}]$
• C. $[\frac{-1}{2},\infty )$
• D. $(\frac{-1}{2},\infty )$

Answer 1

The correct option is A $(-\infty ,\frac{-1}{2})$

Given the function $f\left(x\right)=1-x-x^2$.

Now, $f^{\prime }\left(x\right)=-1-2x$.

Now $f\left(x\right)$ will be increasing if

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

or, $-1-2x\ge 0$

or, $1+2x\le 0$

or, $x\le -\frac{1}{2}$.

So the given function will be decreasing if $-\infty <x\le -\frac{1}{2}$.