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

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 : [0, 1] rarr [0, 1] be a continuous function such that f (f (x))=1 for all x in[0,1] then:

Let f:[0,1]to R be a continuous function then the maximum value of int_(0)^(1)f(x).x^(2)dx-int_(0)^(1)x.(f(x))^(2)dx for all such function(s) is:

Let f(x) be a continuous function such that f(a-x)+f(x)=0 for all x in [0,a] . Then int_0^a dx/(1+e^(f(x)))= (A) a (B) a/2 (C) 1/2f(a) (D) none of these

Let f(x) be a continuous function such that f(a-x)+f(x)=0 for all x in [0,a] . Then, the value of the integral overset(a)underset(0)int (1)/(1+e^(f(x)))dx is equal to

Let f(x) be a continuous function which takes positive values for xge0 and satisfy int_(0)^(x)f(t)dt=x sqrt(f(x)) with f(1)=1/2 . Then

Let f(x) be a continuous function such that f(0) = 1 and f(x)=f(x/7)=x/7 AA x in R, then f(42) is

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

Let f:[0,1]rarrR be a function. Suppose the function f is twice differentiable, f(0)=f(1)=0 and satisfies f\'\'(x)-2f\'(x)+f(x) ge e^x, x in [0,1] Which of the following is true for 0 lt x lt 1 ?