Statement 1: If `f(x)` is differentiable in `[0,1]` such that `f(0)=f(1)=0,` then for any `lambda in R ,` there exists `c` such that `f^prime`(c)`=lambda`f(c),`0ltclt1.` statement 2: if `g(x)` is differentiable in [0,1], where `g(0) =g(1),` then there exists `c` such that `g^prime`(c)=0,

Let f and g be differentiable on [0,1] such that f(0)=2,g(0),f(1)=6 and g(1)=2. Show that there exists c in(0,1) such that f'(c)=2g'(c)

A function f(x) is differentiable at x=c(c in R). Let g(x)=|f(x)|,f(c)=0 then

If f be 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

If f be 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

Let f : R to R be differentiable at c in R and f(c ) = 0 . If g(x) = |f(x) |, then at x = c, g is

A function f is differentiable in the interval 0

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

Statement 1: If both functions f(t)a n dg(t) are continuous on the closed interval [1,b], differentiable on the open interval (a,b) and g^(prime)(t) is not zero on that open interval, then there exists some c in (a , b) such that (f^(prime)(c))/(g^(prime)(c))=(f(b)-f(a))/(g(b)-g(a)) Statement 2: If f(t)a n dg(t) are continuou and differentiable in [a, b], then there exists some c in (a,b) such that f^(prime)(c)=(f(b)-f(a))/(b-a)a n dg^(prime)(c)(g(b)-g(a))/(b-a) from Lagranes mean value theorem.

If f(x) and g(x) ar edifferentiable function for 0lex le1 such that f(0)=2,g(0) = 0,f(1)=6,g(1)=2 , then in the interval (0,1)