Let `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 27 a+9b+3c+d=0, then the equation f(x)=4a x^3+3b x^2+2c x+d=0 has at least one real root lying between (0,3)dot Statement 2: If f(x) is continuous in [a,b], derivable in (a , b) such that f(a)=f(b), then there exists at least one point c in (a , b) such that f^(prime)(c)=0.

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

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 a) cf^(prime)(c)-f(c)=0 b) cf^(prime)(c)+cf(c)=0 c) f^(prime)(c)-cf(c)=0 d) cf^(prime)(c)+f(c)=0

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]for which f^(prime)(c)(f(c))^(n-1)>sqrt(2^(n-1)) , where n in Ndot

Let f be continuous on [a , b],a >0,a n d differentiable on (a , b)dot Prove that there exists c in (a , b) such that (bf(a)-af(b))/(b-a)=f(c)-cf^(prime)(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:Rto R be a twice continuously differentiable function such that f(0)=f(1)=f'(0)=0 . Then

Let f(x)a n dg(x) be differentiable for 0lt=xlt=2 such that f(0)=2,g(0)=1,a n df(2)=8. Let there exist a real number c in [0,2] such that f^(prime)(c)=3g^(prime)(c)dot Then find the value of g(2)dot

Let f(x)a n dg(x) be differentiable for 0lt=xlt=1, such that f(0)=0,g(0)=0,f(1)=6. Let there exists real number c in (0,1) such taht f^(prime)(c)=2g^(prime)(c)dot Then the value of g(1) must be 1 (b) 3 (c) -2 (d) -1

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,