Question

If $f^{\prime }\left(x\right)=g\left(x\right){(x-a)}^2$, where $g\left(a\right)\ne 0$ and is continuous at $x=a,$ then

• A. $f$ is increasing near $a$ if $g\left(a\right)>0$
• B. $f$ is increasing near $a$ if $g\left(a\right)<0$
• C. $f$ is increasing near $a$ if $g\left(a\right)=0$
• D. none of these

Answer 1

Answer: (A)

$f$ is increasing near $a$ if $g\left(a\right)>0$

Since $g$ is continuous at $x=a,$

$\therefore$ if $g\left(a\right)>0,$ then there exists an open interval $I$ containing $a$ so that $g\left(x\right)>0$ for all $x\in I.$

$\Rightarrow f^{\prime }\left(x\right)>0$ for all $x\in I\Rightarrow f$ is increasing near $a$.