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.

 

[tacman]

a) An example where f(x)>0 for all xElement-ofsymbol [a,b] could be the function f(x) = x^2 + 1 on the interval [0,1]. This function is always positive on the interval and satisfies the condition.

An example where f(x)>0 for some xElement-ofsymbol [a,b] could be the function f(x) = sin(x) on the interval [0,π]. This function is positive for some values of x, but also takes on negative values.

b
Integral sign f=0
a

b) If f(x)>0 for all xElement-of symbol[a,b], then this means that the area under the curve is always positive. Since f is continuous on [a,b] and f(x0)>0, we can use the Intermediate Value Theorem to show that there exists a neighborhood of x0 where f(x)>0. This means that the area under the curve on this neighborhood will also be positive. Since the function is continuous, this neighborhood can be extended to cover the entire interval [a,b], thus proving that the integral of f(x) over [a,b] is positive.