Skip to content
dronacharya.ai
← IIT-JEE 2010 · Paper 2
MathematicsInteger+3Question 28 of 57

IIT-JEE 2010 · Paper 2, Mathematics question 28

Let ff be a function defined on R\mathbb R (the set of all real numbers) such that f′(x)=2010(x−2009)(x−2010)2(x−2011)3(x−2012)4f'(x)=2010(x-2009)(x-2010)^2(x-2011)^3(x-2012)^4, for all x∈Rx\in\mathbb R.

If gg is a function defined on R\mathbb R with values in the interval (0,∞)(0,\infty) such that f(x)=ln⁡(g(x)), for all x∈R,f(x)=\ln(g(x)),\ \text{for all } x\in\mathbb R, then the number of points in R\mathbb R at which gg has a local maximum is

Answer and worked solution

Official answer: 1

Same extrema. g=efg=e^{f} and the exponential is increasing, so gg has a local maximum exactly where ff does.

Sign changes of f′f'. The even powers (x−2010)2(x-2010)^2 and (x−2012)4(x-2012)^4 do not change sign. The sign of f′f' is that of (x−2009)(x−2011)3(x-2009)(x-2011)^3:

  • x<2009x<2009: positive; 2009<x<20112009<x<2011: negative; x>2011x>2011: positive.

So ff has a local maximum at 2009 and a local minimum at 2011: one local maximum.

Answer: 1

Topics:Maxima and minima
← Previous questionNext question →