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Question

# Let f(x)=xsinπx,x>0. Then for all natural numbers n, f′(x) vanishes at

A
a unique point in the interval (n,n+12)
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B
a unique point in the interval (n+12,n+1)
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C
a unique point in the interval (n,n+1)
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D
two points in the interval (n,n+1)
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Solution

## The correct option is C a unique point in the interval (n,n+1) f(x)=xsinπx, x>0 f′(x)=sinπx+πxcosπx f′(x)=0⇒tanπx=−πx Clearly, f′(x)=0 has a unique root in the interval (n+12,n+1) Also, f′(x)=0 has a unique root in the interval (n,n+1)

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