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 $P'(x) > 0 \forall \, x \in [0, 1],$ then

• A. $S = \phi$
• 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 $P (x) = ax^2 + bx + c$
Given $P (0) = 0$ and $P (1) = 1 \Rightarrow \, c = 0$ and $a + b = 1$
$\Rightarrow \, a = 1 - b$
$\therefore \, P (x) = (1 - b) x^2 + bx$
$\Rightarrow P' \left(x\right) =2\left(1-b\right)x +b$
Given $ P'\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\} $