|∫f(x)dx|(from a to b=∫|f(x)|(from a to b) then , a

(a) exactly one root in (a, b)

(b) at least one root in (a, b)

(c) no root in (a, b)

(d) None of the above

can someone please help with i

Sorry for answering this question very lately! I’m seeing this question just now!

I think you know that  is positive if and only if the function  lies completely above X-axis in the interval . Similarly, that integral will be negative only if it lies completely below X-axis in .

So, if you have a function like , you can see that it lies above X-axis in and it lies below X-axis in . But, the function  is above X-axis in  and also in . So, now you’ll understand:

 so, here the integrals are not equal

 So,f(x)= sin(x) has one root in 

So, we can conclude that if the integrals are equal, f(x) has no root in (a,b)