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If P(1)=0a n d(d P(x))/(dx),sinx+2xgeq(3...

If `P(1)=0a n d(d P(x))/(dx),sinx+2xgeq(3x(x+1))/pi` . Explain the identity, if any, used in the proof.

Text Solution

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Given , `P(1) =0 and (dP(x))/dx-P(x)gt 0,AAxge1 … (i)`
On Multiplying Eq. (i) by `e^(-x)` we get
`e^(-x) cdot d/dx P(x)cdot d/dx e^(-x)gt0`
`rArr d/dx (P(x) cdot e^(-x))gt0`
`rArr P(x)cdot e^(-x)` is an increasing function.
`rArr P(x)cdot e^(-x)gt P(1) cdot e^(-1), AAxge 1 `
`rArr p(x)gt 0, AAx gt1 [therefore P(1)=0 and e^(-x) gt 0]`
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