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Question

Let f(x) be a polynomial of degree 5 such that x=±1 are its critical points. If limx0(2+f(x)x3)=4, then which of the following is not true ?

A
f(1)4f(1)=4.
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B
x=1 is a point of maxima and x=1 is a point of minimum of f.
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C
f is an odd function.
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D
x=1 is a point of minima and x=1 is a point of maxima of f.
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Solution

The correct option is D x=1 is a point of minima and x=1 is a point of maxima of f.
Given limx0(2+f(x)x3)=4
limx0f(x)x3=2

limx0f(x)x3, Limit exists and it is finite.

f(x)=ax5+bx4+cx3
limx0(ax2+bx+c)=2
c=2

Also, f(x)=5ax4+4bx3+6x2
x=±1 are critical points.
f(1)=5a+4b+6=0
and f(1)=5a4b+6=0
b=0, a=65

f(x)=65x5+2x3
f is odd.
f(x)=6x4+6x2
f′′(x)=24x3+12x
f′′(1)<0 and f′′(1)>0
So, at x=1, there is local minima and at x=1, there is local maxima.

Also, f(1)4f(1)=4

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