Proving f(x)>0 for All xElement-ofsymbol [a,b]

[Swamifez]

If f(x)>0 for all xElement-ofsymbol [a,b], then
b
Integral sign f>0
a

a) Give an example where f(x)>0 for all xElement-ofsymbol [a,b], and f(x)>0 for some xElement-ofsymbol [a,b], and

b
Integral sign f=0
a

b) Suppose that f(x)>0 for all xElement-of symbol[a,b] and f is continuous on at x0 in [a,b] and f(x0)>0. Prove that

b
Integral sign f>0
a

 

[quasar987]

"elements of" is written in words as "in" (ex: for all x in A,...)

As worded, these questions are both equivalent to

"Show that if f f(x)>0 for all xElement-ofsymbol [a,b], then
\int_a^bfdx>0"

But this is precisely the statement of the result you cited in the begining, so the questions are not challenging at all and contain unnecessary words.