Daniel J Smith Custom TMUA Set A Paper 2 · Question 19

Consider the two statements about a real-valued function $f$ :

(A) For all real numbers $x$ there exists a real number $y$ such that $y>x$ and $f(y)<f(x)$

(B) There exists a real number $y$ such that for all real numbers $x$ with $x<y$ , $f(y)<f(x)$

Which of the following is true?

ABoth (A) and (B) are always true for any function

f

BFor some functions

f

, (A) is true and (B) is true, but not for all

f

CThere is a function

f

for which (A) is true and (B) is false, but not the other way round.

DThere is a function

f

for which (B) is true and (A) is false, but not the other way round.

EFor every function

f

, (A) and (B) are either both true or both false.

FThere is no function

f

for which (A) is true.

examday