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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

AI Generated Solution

To solve the problem, we need to find the set of all polynomials \( P(x) \) of degree less than or equal to 2 that satisfy the conditions \( P(1) = 1 \), \( P(0) = 0 \), and \( P'(x) > 0 \) for all \( x \in [0, 1] \). ### Step-by-Step Solution: 1. **Form of the Polynomial**: Since \( P(x) \) is a polynomial of degree less than or equal to 2, we can express it in the form: \[ P(x) = ax^2 + bx + c ...
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