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`

If f is a continuous function on [0,1], differentiable in (0, 1) such that f(1)=0, then there exists some c in (0,1) such that cf^(prime)(c)-f(c)=0 cf^(prime)(c)+cf(c)=0 f^(prime)(c)-cf(c)=0 cf^(prime)(c)+f(c)=0

Leg f be continuous on the interval [0,1] to R such that f(0)=f(1)dot Prove that there exists a point c in [(0,1)/2] such that f(c)=f(c+1/2)dot

Statement-1: If a continuous funtion on [0,1] satisfy 0 le f(x) le 1 , then there exist c in [0,1] such that f(c )=c Statement-2: lim_(x to c) f(x)=f(c)

Let f and g be function continuous in [a,b] and differentiable on [a,b] .If f(a)=f(b)=0 then show that there is a point c in (a,b) such that g'(c) f(c)+f'(c)=0 .

Let fa n dg be differentiable on [0,1] such that f(0)=2,g(0)=0,f(1)=6a n dg(1)=2. Show that there exists c in (0,1) such that f^(prime)(c)=2g^(prime)(c)dot

The function f(x) is defined in [0,1] . Find the domain of f(tanx).

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

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

Let f(x)be continuous on [a,b], differentiable in (a,b) and f(x)ne0"for all"x in[a,b]. Then prove that there exists one c in(a,b)"such that"(f'(c))/(f(c))=(1)/(a-c)+(1)/(b-c).