If the function `f:[0,4]vecR` is differentiable, the show that for `a , b , in (0,4),f^2(4))-(f^2(0))=8f^(prime)(a)f(b)`

Given a function f:[0,4]toR is differentiable ,then prove that for some alpha,beta in (0,2), int_(0)^(4)f(t)dt=2alphaf(alpha^(2))+2betaf(beta^(2)) .

Let f:[2,7]vec[0,oo) be a continuous and differentiable function. Then show that (f(7)-f(2))((f(7))^2+(f(2))^2+f(2)f(7))/3=5f^2(c)f^(prime)(c), where c in [2,7]dot

Let f:[2,7]vec[0,oo) be a continuous and differentiable function. Then show that (f(7)-f(2))((f(7))^2+(f(2))^2+f(2)f(7))/3=5f^2(c)f^(prime)(c), where c in [2,7]dot

A function f: R->R satisfies that equation f(x+y)=f(x)f(y) for all x ,\ y in R , f(x)!=0 . Suppose that the function f(x) is differentiable at x=0 and f^(prime)(0)=2 . Prove that f^(prime)(x)=2\ f(x) .

A function f: R->R satisfies that equation f(x+y)=f(x)f(y) for all x ,\ y in R , f(x)!=0 . Suppose that the function f(x) is differentiable at x=0 and f^(prime)(0)=2 . Prove that f^(prime)(x)=2\ f(x) .

If f(x) is a twice differentiable function such that f(0)=f(1)=f(2)=0 . Then

Let f be a differentiable function such that f'(x) = f(x) + int_(0)^(2) f(x) dx and f(0) = (4-e^(2))/(3) . Find f(x) .

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

Let f: R->R be a differentiable function with f(0)=1 and satisfying the equation f(x+y)=f(x)f^(prime)(y)+f^(prime)(x)f(y) for all x ,\ y in R . Then, the value of (log)_e(f(4)) is _______

Let f: R->R be a differentiable function with f(0)=1 and satisfying the equation f(x+y)=f(x)f^(prime)(y)+f^(prime)(x)f(y) for all x ,\ y in R . Then, the value of (log)_e(f(4)) is _______