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

Question

A polynomial P(x) with real coefficients has the property that P''(x)≠0 for all x Suppose P(0) =1 and P(0)=-1 What can you say about P(1) ? (A) P(1) ge 0 (B) P(1)≠ 0 (C) P(1) le 0 (D) -(1/2)< P(1)< (1/2)

Tardigrade Admin · Accepted Answer

Let P (x) =ax2+bx +c, a ≠0 P' (x) =2ax +b P'' (x) =2a P(0) =c=1 P' (0) =b =-1 ∴ P(x) =ax2-x+1 P(1) =a-1+1=a a ≠0 ∴ P(1)≠0

Q. A polynomial $P (x)$ with real coefficients has the property that $P^{''} (x) \neq = 0$ for all $x$ Suppose $P (0) = 1$ and $P (0) = - 1$ What can you say about $P (1)$ ?

$P (1) \geq 0$

$P (1) \neq = 0$

$P (1) \leq 0$

$- \frac{1}{2} < P (1) < \frac{1}{2}$

Let
$P (x) = a x^{2} + b x + c, a \neq = 0$
$P^{'} (x) = 2 a x + b$
$P^{''} (x) = 2 a$
$P (0) = c = 1$
$P^{'} (0) = b = - 1$
$∴ P (x) = a x^{2} - x + 1$
$P (1) = a - 1 + 1 = a$
$a \neq = 0$
$∴ P (1) \neq = 0$

Q. A polynomial P(x) with real coefficients has the property that P′′(x)=0 for all x Suppose P(0)=1 and P(0)=−1 What can you say about P(1) ?

P(1)≥0

P(1)=0

P(1)≤0

−21​<P(1)<21​