which of the following statement is true/false?

Question

Given $f : [ \frac{1}{2}, 2] \rightarrow \mathbb{R} $ , a strictly increasing function , we put $g(x) = f(x) + f(1/x) , x \in [1,2]$. Consider a partition $P$ of $[1, 2] $ and let $U(P,g)$ and $L(P, g)$ denotes the upper sum Reimann sum and lower Reimann sum of $g$ . Then which of the following statement is true/false ?

$1.$ For a suitable $f$ we can have $U(P, g) = L (P, g)$

$2$. for a suitable $f$ we can have $U(P, g) \neq L (P,g)$

$3$.$U(P,g) \ge L (P,g)$ for all choices of $f$

$4$. $U(P,g) < L(P,g) $for all choices of $f$

any hints/solution

Answer 1

1. This is correct. Take $f(x)=\log(x)$. Then $g(x)=0$.
2. This is correct too. Take $f(x)=x$, for instance.
3. This is true, of course.
4. This inequality never occurs.