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`

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 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)

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 be continuous on [a,b],a>0, and differentiable on (a,b). Prove that there exists c in(a,b) such that (bf(a,b))/(b-a)=f(c)-cf'(c)

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))

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

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)

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.