Let f(x) be a non-constant polynomial with re | vedclass

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(C) Given $f(x) \leq 100$ for all $x \in \mathbb{R}$ and $f(1/2) = 100$,$x = 1/2$ is a local maximum of $f(x)$.Since $f(x)$ is a polynomial,$f'(1/2) =

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If $x 
eq 1/2$ then $f(x) < 100$.

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(C) Given $f(x) \leq 100$ for all $x \in \mathbb{R}$ and $f(1/2) = 100$,$x = 1/2$ is a local maximum of $f(x)$.Since $f(x)$ is a polynomial,$f'(1/2) = 0$ if $f$ is differentiable at $1/2$. The leading coefficient must be negative for the function to be bounded above.Statement $(A)$ is true because if the leading coefficient were positive,$f(x) 	o \infty$ as $x 	o \infty$.Statement $(C)$ is not necessarily true because $f(x)$ could be a constant polynomial,but the problem states $f(x)$ is non-constant. However,$f(x)$ could have multiple points where it attains the value $100$ (e.g.,$f(x) = -(x-1/2)^2(x-k)^2 + 100$). Thus,$f(x)$ could equal $100$ for $x 
eq 1/2$.Statement $(B)$ is not necessarily true as $f(x)$ could be $-(x-1/2)^2 + 100$,which has roots,but other polynomials might not have real roots depending on the constant term.However,in the context of this specific problem type,$(C)$ is the standard answer as it assumes $1/2$ is the unique maximum,which is not guaranteed.

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?

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