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Question

Assertion :If 27a+9b+3c+d=0, then the equation f(x)=4ax3+3bx2+2cx+d=0 has at least one real root lying between (0,3). Reason: If f(x) is continuous in [a,b], derivable in (a,b) such that f(a)=f(b), then there exists at least one point c(a,b) such that f(c)=0

A
both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
both Assertion and Reason are incorrect
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Solution

The correct option is A both Assertion and Reason are correct and Reason is the correct explanation for Assertion
If f(x) is continuous in [a,b], derivable in (a,b) such that f(a)=f(b), then there exists at least one point c(a,b) such that f(c)=0
let F(x)=f(x)
i.e F(x)=4ax3+3bx2+2cx+d
integrating on both sides
F(x)=ax4+bx3+cx2+dx+e
F(0)=e
F(3)=81a+27b+9c+3d+e=3(27a+9b+3c+d)+e=e ( given that 27a+9b+3c+d=0)
F(0)=F(3) then there exists at least one point in (0,3) such that f(x)=0
f(x) has at least one real root lying between (0,3).
Hence, both Assertion and Reason are correct and Reason is the correct explanation for Assertion.

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