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Let $f(x)$ be a non-constant polynomial with real coefficients such that $f\left(\frac{1}{2}\right)=100$ and $f(x) \leq 100$ for all real $x$. Which of the following statements is NOT necessarily true?
The coefficient of the highest degree term in $f(x)$ is negative.
$f(x)$ has at least two real roots.
If $x \neq 1 / 2$ then $f(x) < 100$.
At least one of the coefficients of $f(x)$ is bigger than $50.$
Solution
(c)
We have, $f\left(\frac{1}{2}\right)=100$
$f(x) \leq 100, \forall x \in R$
$\therefore \quad f(x)=a\left(x-\frac{1}{2}\right)$
$\left[a_0 x^{n-1}+a_1 x^{n-2}+\ldots+a_{n-1}\right]+100$
If $f(x) \leq 100, \forall x \in R$
$\therefore a < 0$ and $f(x)$ must be even degree polynomial.
Since, there may be more value of $x$ at which $f(x)$ attains maximum.
$\therefore$ If $x \neq \frac{1}{2}$, then $f(x) < 100$ may be false.
