TMUA Official 2021 Paper 2 · Question 9 · ExamDay

Official papers

Official 2021 Paper 2 · Q9

Paper 2Logic & proofAlgebra

Consider the following statements about a polynomial $f(x)$ :

I. $f(x)=px^3+qx^2+rx+s$ , where $p\ne0$ .
II. There is a real number $t$ for which $f'(t)=0$ .
III. There are real numbers $u$ and $v$ for which $f(u)f(v)<0$ .

Which of these statements is/are sufficient for the equation $f(x)=0$ to have a real solution?

	Statement I is sufficient	Statement II is sufficient	Statement III is sufficient
A	Yes	Yes	Yes
B	Yes	Yes	No
C	Yes	No	Yes
D	Yes	No	No
E	No	Yes	Yes
F	No	Yes	No
G	No	No	Yes
H	No	No	No

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