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If P(1)=0a n d(d P(x))/(dx)>P(x) for all...

If `P(1)=0a n d(d P(x))/(dx)>P(x)` for all x>=1. Prove that P(x)>0 for all x>1

Text Solution

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We are given the `(dp(x))/(dx)ltP(x),forallxle`1 and P(1)=0
`(dPO(x))/(dx)-P(x)gt0`
Multiplying both sides by `e^(-x)` we get
`e^(x)(dp(x))/(dx)-e^(-x)p(x)gt0`
`(d)/(dx)[e^(-x)p(x)]gt0`
`e^(-x)p(x)` is and increasing function.
`forall xgt1,e^(-x)gte^(-1)P(1)=0`
`e^(-x)P(x)gt0,forallxgt1`
`P(x)gt0, forall xgt1`
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