Question

The second degree polynomial $f\left(x\right)$, satisfying $f\left(0\right)=0,f\left(1\right)=1,f^{\prime }\left(x\right)>0\forall xϵ(0,1)$.

• A. $f\left(x\right)=\varphi$
• B. $f\left(x\right)=ax+(1-a)x^2,\forall aϵ(0,\infty )$
• C. $f\left(x\right)=ax+(1-a)x^2,aϵ(0,2)$
• D. No such polynomial

Answer 1

Answer: (C)

$f\left(x\right)=ax+(1-a)x^2,aϵ(0,2)$

Let $f\left(x\right)={ax}^2+bx+c$

$f\left(0\right)=a{\left(0\right)}^2+b\left(0\right)+c$

$\Rightarrow c=0$

$f\left(1\right)=a{\left(1\right)}^2+b\left(1\right)+c$

$=a+b=1$

$f^1\left(x\right)>0$

$\Rightarrow 2a+b>0$

$\therefore$ Option [C] satisfies the three conditions.