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: (-2, 2) rarr (-2, 2) be a continuous function such that f(x) = f(x^2) AA Χin d_f, and f(0) = 1/2 , then the value of 4f(1/4) is equal to

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)

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:[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 ?

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 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(0) = 1 and f(x)=f(x/7)=x/7 AA x in R, then f(42) is

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:[0,1] rarr R be a function.such that f(0)=f(1)=0 and f''(x)+f(x) ge e^x for all x in [0,1] .If the fucntion f(x)e^(-x) assumes its minimum in the interval [0,1] at x=1/4 which of the following is true ?