Let be a continuous function on the interval If and ,then prove that must have atleast two real roots on the interval
Sorry,I could not update earlier.The solution is a joy to share.I found it on another website though but it doesn't really matter.One can't claim intellectual propriety all the time.So here it is.
Assume is identically not zero,else the result is trivial.
Since on and ,
this leads to the conclusion that takes on positive and negative values in this interval and therefore due to Intermediate Value theorem, has atleast one root in the open interval
Suppose now has exactly one root in this interval say .
Without loss of generality,we can say that for and for
Now consider
Clearly, .
For and for .
For
and thus is strictly positive
which is a contradiction since
Thus our assumption that has exactly one root on is false.
has atleast two roots on