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|∫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

|∫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

Grade:12th pass

1 Answers

Sai Ram Charan
31 Points
5 years ago
Sorry for answering this question very lately! I’m seeing this question just now!
I think you know that \int_{a}^{b}f(x)dx is positive if and only if the function y=f(x) lies completely above X-axis in the interval(a,b) . Similarly, that integral will be negative only if it lies completely below X-axis in (a,b).
So, if you have a function like y=\sin (x), you can see that it lies above X-axis in(0,\pi) and it lies below X-axis in (\pi,2\pi). But, the function y=\left | \sin (x) \right | is above X-axis in (0,\pi) and also in (\pi,2\pi). So, now you’ll understand:
\int_{0}^{2\pi}\left | \sin (x)\right |dx =4 \textup{ but} \left | \int_{0}^{2\pi}\sin (x)dx \right |=0 so, here the integrals are not equal
 So,f(x)= sin(x) has one root in (0,\pi)
So, we can conclude that if the integrals are equal, f(x) has no root in (a,b)
 

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