TMUA Official 2021 Paper 2 · Question 8

Consider the following statement about the polynomial $p(x)$ , where $a$ and $b$ are real numbers with $a<b$ :

$(*)$ There exists a number $c$ with $a<c<b$ such that $p'(c)=0$ .

Which one of the following is true?

AThe condition

p(a)=p(b)

is necessary and sufficient for

(*)

BThe condition

p(a)=p(b)

is necessary but not sufficient for

(*)

CThe condition

p(a)=p(b)

is sufficient but not necessary for

(*)

DThe condition

p(a)=p(b)

is not necessary and not sufficient for

(*)

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