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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 options are B a unique point in the interval (n+12, n+1) C a unique point in the interval (n, n+1)Given that f(x)=xsinπx⇒f′(x)=sinπx+πxcosπxf′(x)=0⇒sinπx+πxcosπx=0⇒tanπx=−πx⇒πx∈ (2n+12π, (n+1)π)⇒x∈(n+12, n+1) and also x∈(n, n+1).

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