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Let S denote the set of all polynomia...

Let `S` denote the set of all polynomials `P(x)` of degree `lt=2` such that `P(1)=1,P(0)=0a n dP^(prime)(x)>0AAx in [0,1]` , then `S=varphi` b. `S={(1-a)x^2+a x ;0

A

`S=0`

B

`S=ax+(1-a)x^(2),AAa epsilon(0,oo)`

C

`S=ax+(1-a)x^(2),AA a epsilonR`

D

`S=ax+(1-a)x^(2),AA a epsilon(0,2)`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `P(x)=bx^(2)+ax+c`
`impliesc=0`
AS `P(1)=1`
`P(x)=ax+(1-a)x^(2)`
Now `P'(x)=a+2(1-a)x`
As `P'(x)gt0` for `x epsioon (0,1)`
Only option d satisfies above condition.
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