`f: [0.4]-> R` is a differentiable function. Then prove that for some `a,bin(0,4)`, `f^2(4)-f^2(0)=8f'(a)*f(b)`.

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

Let f(x) be a differentiable function in the interval (0, 2) then the value of int_(0)^(2)f(x)dx

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

Let f"[2,7]rarr[0,oo) be a continuous and differentiable function. Then, the value of (f(7)-f(2))((f(7))^(2)+(f(2))^(2)+f(2).f(7))/(3) is (where c in (2,7) )

If f: R -> R and g: R ->R are two given functions, then prove that 2min {f(x)-g(x),0}=f(x)-g(x)-|g(x)-f(x)|

If f(x) is a twice differentiable function such that f(a)=0, f(b)=2, f(c)=-1,f(d)=2, f(e)=0 where a < b < c < d e, then the minimum number of zeroes of g(x) = f'(x)^2+f''(x)f(x) in the interval [a, e] is

If f: [-5, 5] rarr R is a differentiable function and if f'(x) does not vanish anywhere, then prove that f(-5) ne f(5)

f(x) is a continuous and differentiable function. f'(x) ne 0 and |{:(f'(x),f(x)),(f''(x),f'(x)):}|=0 . If f(0)=1 and f'(0)=2 then f(x)=....

Let f be an even function and f'(0) exists, then f'(0) is

Let y = f(x), f : R ->R be an odd differentiable function such that f'''(x)>0 and g(alpha,beta)=sin^8alpha+cos^8beta+2-4sin^2alpha cos^2 beta If f''(g(alpha, beta))=0 then sin^2alpha+sin^2beta is equal to