if f(x) has domain [0-1] and range [0-1] and iscontinuos n differneitable show dat for sum x...f(x)=x
66If f(0)=0 or f(1)=1, we are done. So WLOG, lets assume that f(0)>0 and f(1)<1. Now, consider the function
g(x) = f(x) - x.
Obviously, g is continuous in the interval [0,1]. Further,
g(0)=f(0)-0 >0 and
g(1)=f(1)-1 <0
Hence, by intermediate value theorem, there exists some x in the interval (0,1) such that g(x)=0 which means that f(x)=x.
24how did u assume f(0)>0 and f(1)<1
and g(x) = f(x) - x.
I mean why this only and not something else........
anyways soln is rally good
24the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is a corresponding value in its domain mapping to the original.
1Actually Eureka , the max value of f(x) is 1 .... f(1)-1 can be equal to or less than 0. If it's equal then we have already found a solution ... so we assume it 's less than 0.same for the other case... @ rohit As for intermediate value theorem ref. to calculus by Maron... it's very handy....
