Question

If P(x) is a polynomial of degree less than or equal to 2 and S is the set of all such polynomials so that $P(0)=0,P(1)=1$ and $P^{\prime }(x)>0\forall x\in [0,1],$ then

• A. $S=\varphi$
• B. $S=ax+(1-a)x^2\forall a\in (0,2)$
• C. $S=ax+(1-a)x^2\forall a\in (0,\infty )$
• D. $S=ax+(1-a)x^2\forall a\in (0,1)$

Answer 1

Answer: (B)

Let the polynominal be $P(x)=a{x}^2+bx+c$
Given $P(0)=0$ and $P(1)=1\Rightarrow c=0$ and $a+b=1$
$\Rightarrow a=1-b$
$\therefore P(x)=(1-b)x^2+bx$
$\Rightarrow P^{\prime }\left(x\right)=2\left(1-b\right)x+b$
Given $P^{\prime }\left(x\right)>0,\forall x\in \left[0,1\right]$
$\Rightarrow 2\left(1-b\right)x+b>0$
$\Rightarrow$ When $x=0,b>0$ and when $x=1,b>2$
$\Rightarrow 0<b<2$
$\therefore S=\left\{\left(1-a\right)x^2+ax,a\in \left(0,2\right)\right\}$