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Question

Let f(x)=x44x3+4x2+c, cR. Then

A
f(x) has infinitely many zeros in (1,2) for all c
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B
f(x) has exactly one zero in (1,2) if 1<c<0
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C
f(x) has double zeros in (1,2) if 1<c<0
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D
whatever be the value of c,f(x) has no zero in (1,2)
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Solution

The correct option is B f(x) has exactly one zero in (1,2) if 1<c<0
f(x)=x44x3+4x2+c
f(x)=4x312x2+8x
=4x(x1)(x2)
Critical points are : 0,1,2

For interval (1,2),
f(1)=1+c, f(2)=c

f(1)f(2)<0
(1+c)c<0
1<c<0
f(x) has exactly one zero in (1,2) if 1<c<0

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