Application of Derivatives 4 Question 64
67. Let $f$ be a function defined on $R$ (the set of all real numbers) such that $f^{\prime}(x)=2010(x-2009)(x-2010)^{2}$ $(x-2011)^{3}(x-2012)^{4}, \forall x \in R$. If $g$ is a function defined on $R$ with values in the interval $(0, \infty)$ such that $f(x)=\ln (g(x)), \forall x \in R$, then the number of points in $R$ at which $g$ has a local maximum is……
(2010)
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Solution:
- 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.
