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

Question

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

Aicharya · Accepted Answer

Let $g(x)=e^{f(x)}, \forall x \in R$ $ \Rightarrow g^{\prime}(x)=e^{f(x)} \cdot f^{\prime}(x) $ $\Rightarrow f^{\prime}(x)$ changes its sign from positive to negative in the neighbourhood of $x=2009$ $\Rightarrow f(x)$ has local maxima at $x=2009$ So, the number of local maximum is one

Let $f$ be a function defined on $R$ (the set of all real numbers) such that $f^{\prime}(x)=2010(x-2009)$…

Solution