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

Options

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)`

Answer

D

Explanation

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.