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Question

Let then f(x)=(x4)(x5)(x6)(x7) then

A
f(x)=0 has four real roots
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B
three roots of f(x)=0 lie in (4,5)(5,6)(6,7)
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C
the euation f(x)= has only two roots>d. three roots of f(x)=0(3,4)(4,5)(5,6)
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D
three roots of f(x)=0 (3,4)(4,5)(5,6)
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Solution

The correct option is B three roots of f(x)=0 lie in (4,5)(5,6)(6,7)
Since f(4)=f(5)=f(6)=f(7)=0
So by Rolle's theorem applied on the interval [4,5],[5,6],[6,7]
There are x1(4,5),x2(5,6),x3(6,7)
Such that f(x1)=f(x2)=f(x3)=0.
Since f is polynomial of degree 3 so cannot have four roots
Hence f(x) has 3 roots in (4,5)(5,6)(6,7)

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