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Let $\mathrm{f}(\mathrm{x})$ be a non-constant polynomial with real coefficients such that $\mathrm{f}\left(\frac{1}{2}\right)=100$ and $\mathrm{f}(\mathrm{x}) \leq 100$ for all real $\mathrm{x}$. Which of the following statements is NOT necessarily true?
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The correct answer is:
If $x \neq 1 / 2$ then $f(x) < 100$
Coefficient of highest degree term must be negative becuase ifit is positive, then $\mathrm{x} \rightarrow \infty, \mathrm{y} \rightarrow \infty$ and it is not possible, since $f(x) \leq 100$.
Now, graph will be like
at least two real roots will be there, \& if $x \neq \frac{1}{2}$, then $f(x) < 100$, it is not always true, as the graph can be like. this also
Now, let the highest coefficients, it can have is 49
then, $f\left(\frac{1}{2}\right)=49+\frac{49}{2}+\frac{49}{2^{2}}+\ldots$
But the sum cannot be equal to 100 .
Now, graph will be like
at least two real roots will be there, \& if $x \neq \frac{1}{2}$, then $f(x) < 100$, it is not always true, as the graph can be like. this also
Now, let the highest coefficients, it can have is 49
then, $f\left(\frac{1}{2}\right)=49+\frac{49}{2}+\frac{49}{2^{2}}+\ldots$
But the sum cannot be equal to 100 .
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