Question

A polynomial $P(x)$ with real coefficients has the property that $P''\left(x\right)\ne0$ for all $x$ Suppose $P(0) =1$ and $P(0)=-1$ What can you say about $P(1)$ ?

• A. $P\left(1\right)\ge\,0$
• B. $P\left(1\right)\ne\,0$
• C. $P\left(1\right)\le\,0$
• D. $-\frac{1}{2}<\,P\left(1\right)<\,\frac{1}{2}$

Answer 1

Answer: (B)

Let
$P (x) =ax^{2}+bx +c, a \ne0$
$P' (x) =2ax +b$
$P'' (x) =2a$
$P(0) =c=1$
$P' (0) =b =-1$
$\therefore P(x) =ax^{2}-x+1$
$P(1) =a-1+1=a$
$a \ne0$
$\therefore P\left(1\right)\ne0$