Consider the following statement about the polynomial $\mathrm { p } ( x )$, where $a$ and $b$ are real numbers with $a < b$ : (*) There exists a number $c$ with $a < c < b$ such that $\mathrm { p } ^ { \prime } ( c ) = 0$. Which one of the following is true? A The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is necessary and sufficient for ( $*$ ) B The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is necessary but not sufficient for (*) C The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is sufficient but not necessary for ( $*$ ) D The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is not necessary and not sufficient for ( $*$ )
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Consider the following statement about the polynomial $\mathrm { p } ( x )$, where $a$ and $b$ are real numbers with $a < b$ :\\
(*) There exists a number $c$ with $a < c < b$ such that $\mathrm { p } ^ { \prime } ( c ) = 0$.
Which one of the following is true?
A The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is necessary and sufficient for ( $*$ )\\
B The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is necessary but not sufficient for (*)\\
C The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is sufficient but not necessary for ( $*$ )\\
D The condition $\mathrm { p } ( a ) = \mathrm { p } ( b )$ is not necessary and not sufficient for ( $*$ )