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A polynomial P(x) with real coefficients...

A polynomial P(x) with real coefficients has the property that `P^(n)(x)ne0` for all x. Suppose `P(0)=1` and P'(0) = `-1`.
What can you say about P(1)?

A

`P(1)ge0`

B

`P(1)ne0`

C

`P(1)le0`

D

`-1//2ltP(1)lt1//2`

Text Solution

Verified by Experts

The correct Answer is:
C
`P(x)=e^(-x)" "P(0)=1`
`P'(x)=-e^(-x)" "P'(0)=-1" "P''(x)=e^(-x)ne0AAx in 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" "P(1)ne0`
`=-0.7`
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