IIT-JEE 2005 ScreeningQuestion 1
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 $P'(x)>0$ for all $x\in[0,1]$, then
Differentiation and its Application
Solution
Concept. $P'$ is linear, so it is positive on $[0,1]$ exactly when it is positive at both ends. Formulas used. (1) A linear function is positive on $[0,1]$ iff it is positive at $0$ and at $1$Step 1 - Form. $P(0)=0$ and $P(1)=1$: $P=bx+(1-b)x^2$. Step 2 - Derivative. $P'(x)=b+2(1-b)x$, so $P'(0)=b$ and $P'(1)=2-b$. Step 3 - Condition. By (1), $b>0$ and $2-b>0$: $0<b<2$. Step 4 - Set. $S=\{ax+(1-a)x^2: a\in(0,2)\}$. Answer: (B)