JEE Advance - Mathematics (2010 - Paper 2 Offline - No. 10)
Let $$f$$ be a function defined on $$R$$ (the set of all real numbers)
such that $$f'\left( x \right) = 2010\left( {x - 2009} \right){\left( {x - 2010} \right)^2}{\left( {x - 2011} \right)^3}{\left( {x - 2012} \right)^4}$$ for all $$x \in $$$$R$$
such that $$f'\left( x \right) = 2010\left( {x - 2009} \right){\left( {x - 2010} \right)^2}{\left( {x - 2011} \right)^3}{\left( {x - 2012} \right)^4}$$ for all $$x \in $$$$R$$
If $$g$$ is a function defined on $$R$$ with values in the interval $$\left( {0,\infty } \right)$$ such that
$$$f\left( x \right) = ln\,\left( {g\left( x \right)} \right),\,\,for\,\,all\,\,x \in R$$$
then the number of points in $$R$$ at which $$g$$ has a local maximum is ___________.
Answer
1
Explanation
Let $$g(x) = {e^{f(x)}},\,\forall x \in R$$
$$ \Rightarrow g'(x) = {e^{f(x)}}\,.\,f'(x)$$
$$\Rightarrow$$ f'(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.
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