If S is a set of polynomials P(x) of degree l | vedclass

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(D) Let $P(x) = bx^2 + ax + c$.Given $P(0) = 0$,we have $c = 0$.Given $P(1) = 1$,we have $a + b = 1$,which implies $b = 1 - a$.Thus,$P(x) = (1 - a)x^2

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$S = \{ax + (1 - a)x^2 : a \in (0, 2)\}$

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(D) Let $P(x) = bx^2 + ax + c$.Given $P(0) = 0$,we have $c = 0$.Given $P(1) = 1$,we have $a + b = 1$,which implies $b = 1 - a$.Thus,$P(x) = (1 - a)x^2 + ax$.The derivative is $P'(x) = 2(1 - a)x + a$.We are given $P'(x) > 0$ for all $x \in (0, 1)$.At $x = 0$,$P'(0) = a > 0$.At $x = 1$,$P'(1) = 2(1 - a) + a = 2 - a > 0$,which implies $a Since $P'(x)$ is a linear function,if it is positive at the endpoints of the interval $(0, 1)$,it is positive for all $x \in (0, 1)$.Thus,$0 Therefore,$S = \{ax + (1 - a)x^2 : a \in (0, 2)\}$.

If $S$ is a set of polynomials $P(x)$ of degree $\le 2$ such that $P(0) = 0$,$P(1) = 1$,and $P'(x) > 0$ for all $x \in (0, 1)$,then which of the following describes $S$?

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