Question

Let $f\left(x\right)$ be a non-constant polynomial with real coefficients such that $f\left(\frac{1}{2}\right)=100$ and $f\left(x\right)\le 100$ for all real $x$. Which of the following statements is NOT necessarily true?

• A. The coefficients of the highest degree term in $f\left(x\right)$ is negative
• B. $f\left(x\right)$ has at least two real roots
• C. If $x\ne 1/2$ then $f\left(x\right)<100$
• D. At least one of the coefficients of $f\left(x\right)$ is bigger than $50$

Answer 1

Answer: (C)

If $x\ne 1/2$ then $f\left(x\right)<100$

Coefficient of of highest degree term must be negative because if it is positive, then $x\to \infty ,y\to \infty$ and it is not possible, since f(x) $\le$ 100.

$\therefore$ option (A) is correct.

as $x\to \infty$, $f\left(x\right)\to -\infty$ and at $x=1/2$, $f\left(x\right)=100$ and as $x\to -\infty ,f\left(x\right)\to -\infty$

$\therefore$ there exist at least one root in $(-\infty ,1/2)$, and $(1/2,\infty )$

$\therefore$ option (B) is correct.

consider the function $f\left(x\right)=100-(x-1/2{)}^2(x+1/2{)}^2$

observe that $f\left(1/2\right)=100$ and $f(-1/2)=100$

$\therefore$ option (C) need not be correct

Now, let the highest coefficients, it can have is 49.

then, $f\left(1/2\right)=49+\frac{49}{2^2}+\frac{49}{2^3}+...$

but the sum cannot be equal to $100$

$\therefore$ there should be at least one coefficient greater than 50 in order to satisfy the above conditions.

$\therefore$ option (D) is correct.