Points of Maxima

Calculus Level 4

Let $f(x)$ be a function defined on $\mathbb R$ (the set of all real numbers) such that $f'(x)=2010 (x-2009)(x-2010)^2(x-2011)^3(x-2014)^4$ for all $x\in\mathbb R$ .

If $g(x)$ is a function defined on $\mathbb R$ with values in the interval $(0,\infty)$ such that $f(x)=\ln (g(x))$ for all $x\in \mathbb R$ , then find the number of points in $\mathbb R$ at which $g(x)$ has a local maximum.

The answer is 1.

1 solution

Otto Bretscher
May 2, 2016

Great calculus problem! I will use it in an exam sometime soon. Thanks, Comrade!

$f'(x)$ has two sign changes, from positive to negative at 2009 and from negative to positive at 2011. Thus $f(x)$ has a local maximum at 2009 and a local minimum at 2011. The extrema of $g(x)=e^{f(x)}$ are attained at the same points since $e^t$ is increasing, so that $g(x)$ has $\boxed{1}$ local maximum, attained at $x=2009$ .

This question has previously appeared in a IIT JEE Entrance. The exam is known for such problems :-)

Pulkit Gupta - 5 years, 1 month ago

JEE 2010 question

Arunava Das - 3 years, 4 months ago

Points of Maxima

The answer is 1.

1 solution

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