A polynomial P(x) with real coefficients has the property that P "(x ) \neq 0 for all x. Suppose P(0) = 1 and P '{0) = -1. What can you say about P( 1)?

KVPY SX SB Exam 2012 Question Paper

Question : 85 of 120

Marks: +1, -0

A polynomial P(x) with real coefficients has the property that

P "(x ) ≠ 0

for all x. Suppose

P(0) = 1

and

P '{0) = -1

. What can you say about

P( 1)

Solution:

P(x)=e^{-x},P(0)=1

P'(x)=-e^{-x},P'(0)=-1

P"(x)=e^{-x}≠0∀x∈R

P(1)=1/e

P(x)=-e^x+2

P'(x)=-e^x

P'(0)=-1

P"(x)=-e^x

P(1)=-e+2=-0.7

(P(1)≠0)

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