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 g(x) be the inverse of an invertible function f(x), which is differentiable for all real xdot Then g^(f(x)) equals. -(f^(x))/((f^'(x))^3) (b) (f^(prime)(x)f^(x)-(f^(prime)(x))^3)/(f^(prime)(x)) (f^(prime)(x)f^(x)-(f^(prime)(x))^2)/((f^(prime)(x))^2) (d) none of these

Let f:[2,7]to[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 (x) be twice differentiable function such that f'' (x) lt 0 in [0,2]. 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) .

Let f:[0,2]vecR be a function which is continuous on [0,2] and is differentiable on (0,2) with f(0)=1 L e t :F(x)=int_0^(x^2)f(sqrt(t))dt for x in [0,2] . If F^(prime)(x)=f^(prime)(x) . for all x in (0,2), then F(2) equals

For the function f, f(x)=x^2-4x+7 , show that f '(5)= 2f'(7/2) .

If f is an odd continuous function in [-1,1] and differentiable in (-1,1) then a. f'(A)=f(1)" for some "A in (-1,0) b. f'(B) =f(1)" for some "B in (0,1) c. n(f(A))^(n-1)f'(A)=(f(1))^(n)" for some " A in (-1, 0), n in N d. n(f(B))^(n-1)f'(B)=(f(1))^(n)" for some" B in (0,1), n in N

Fill in the blanks: If a function f is the differentiable at c such that f' (c ) = 0 and f" (c ) < 0, then c is a point of ________.

If f and g are continuous functions on [0,a] satisfying f(x)=(a-x) and g(x)+g(a-x)=2 , then show that int_(0)^(a)f(x)g(x)dx=int_(0)^(a)f(x)dx .