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`

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(x) is continuous in [a , b] and differentiable in (a,b), then prove that there exists at least one c in (a , b) such that (f^(prime)(c))/(3c^2)=(f(b)-f(a))/(b^3-a^3)

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 be a continuous, differentiable, and bijective function. If the tangent to y=f(x) a tx=a is also the normal to y=f(x) a tx=b , then there exists at least one c in (a , b) such that (a)f^(prime)(c)=0 (b) f^(prime)(c)>0 (c) f^(prime)(c)<0 (d) none of these

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

If f(x)a n dg(x) are continuous functions in [a , b] and are differentiable in (a , b) then prove that there exists at least one c in (a , b) for which. |f(a)f(b)g(a)g(b)|=(b-a)|f(a)f^(prime)(c)g(a)g^(prime)(c)|, w h e r e a

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

Statement 1: If both functions f(t)a n dg(t) are continuous on the closed interval [a,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 continuous 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 Lagranges mean value theorem.

Let f:Rto R be a twice continuously differentiable function such that f(0)=f(1)=f'(0)=0 . Then