Let `f:[0,1] rarr [0,1]` be a continuous function. Then prove that `f(x)=x` for at least one `0lt=xlt=1.`

Let f: R rarr (0,1) be a continuous function. Then, which of the following function (s) has (have) the value zero at some point in the interval (0,1)?

Let f,g:[-1,2]rarr RR be continuous functions which are twice differentiable on the interval (-1, 2). Let the values of f and g at the points –1, 0 and 2 be as given in the following table : x=-1 x=0 x=2 f(x) 3 6 0 g(x) 0 1 -1 In each of the intervals (-1,0) and (0, 2) the function (f – 3g)'' never vanishes. Then the correct statement(s) is(are)

A continuous function f(x) satisfies the relation f(x)=e^x+int_0^1 e^xf(t)dt then f(1)=

Let f: RvecR be a continuous function which satisfies f(x)= int_0^xf(t)dtdot Then the value of f(1n5) is______

If f is continuous and differentiable function and f(0)=1,f(1)=2, then prove that there exists at least one c in [0,1]forw h i c hf^(prime)(c)(f(c))^(n-1)>sqrt(2^(n-1)) , where n in Ndot

Let f: R->R be a continuous function and f(x)=f(2x) is true AAx in Rdot If f(1)=3, then the value of int_(-1)^1f(f(x))dx is equal to (a)6 (b) 0 (c) 3f(3) (d) 2f(0)

Let f be a continuous function satisfying f '(I n x)=[1 for 0 1 and f (0) = 0 then f(x) can be defined as

Let f be a continuous function defined onto [0,1] with range [0,1]. Show that there is some c in [0,1] such that f(c)=1-c

Let f:(0,oo)->R be a differentiable function such that f'(x)=2-f(x)/x for all x in (0,oo) and f(1)=1 , then

Discuss the continuity of the function f defined by f(x)= (1)/(x), x ne 0 .