Question

Let $f$ be a function defined on $R$ (the set of all real numbers) such that $f^{\prime }\left(x\right)=2010(x-2009)(x-2010{)}^2(x-2011{)}^3(x-2012{)}^4$, for all x $\in R$. If $g$ is a function defined on $R$ with values in the interval $(0,\infty )$ such that $f\left(x\right)=\ln \left(g\right(x\left)\right)$ , for all $x\in R$, then the number of points in $R$ at which $g$ has a local maximum is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is B 1

$f\left(x\right)=\ln \left\{g\right(x\left)\right\}$

$g\left(x\right)=e^{f\left(x\right)}$

$g^{\prime }\left(x\right)=e^{f\left(x\right)}.f^{\prime }\left(x\right)$

$g^{\prime }\left(x\right)=0\Rightarrow f^{\prime }\left(x\right)=0$ as

$e^{f\left(x\right)}\ne 0$

$\Rightarrow 2010(x-2009)(x-2010{)}^2(x-2011{)}^3(x-2012{)}^4=0$

so there is only one point of local maxima.