Question

If $f^{\prime }\left(x\right)=g\left(x\right){(x-a)}^2$, where $g\left(a\right)\ne 0$ and $g$ 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 decreasing near a if $g\left(a\right)>0$
• D. $f$ is decreasing near a if $g\left(a\right)<0$

Answer 1

Answer: (A), (D)

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

Since $g$ is continuous at $x=a$
if $g\left(a\right)>0$, there exist an open interval $I$ containing $a$ so that $g\left(x\right)>0\forall x\in I$
$\Rightarrow f^{\prime }\left(x\right)\ge 0\forall x\in I$. Therefore, $f$ is increasing near $a$.
Similarly $f$ is decreasing near $a$ if $g\left(a\right)<0$.
Ans: A,D