Let $f$ be a strictly monotonic continous real valued function defined on $[a,b]$ such that $f(a)<a$ and $f(b)>b$. Then, which one of the following is true?
a) There exists exactly one $c \in (a,b)$ such that $f(c)=c$
b) There exist exactly two points $c_1,c_2 \in (a,b)$ such that $f(c_i)=c_i$ for $i=1,2$
c)There is no $c \in (a,b)$ such that $f(c)=c$
d)There exist infinitely many points $c \in (a,b)$ such that $f(c)=c$
given: $f(a)<a$ and $f(b)>b$
implies $f(-a)>-a$
hence $f(b)-f(a)>b-a \implies\frac {f(b)-f(a)} {b-a} \gt {1}$
By Lagrange Mean value theorem there exist $c\in(a,b)$ such that $\frac {f(b)-f(a)} {b-a} =f'(c)$
hence $f'(c)>1$ that is given function is strictly increasing at $c$.
so what we can conclude from this?