Let f is continuous on [a, b] and differentiable on `(a,b)s.t.t^2(a)-t^2(b)=a^2-b^2.` Show that ...`f(x) f prime (x) = x ` has atleast one root in `(a, b).`

Let f is continuous on [a,b] and differentiable on (a,b)s.t.t^(2)(a)-t^(2)(b)=a^(2)-b^(2) .Show that....f(x)f^(2)(x)=x has atleast one root in (a,b).

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)

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)

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)

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)

If f(x) is continuous in [a, b] and differentiable in (a, b), prove that there is atleast one c in (a, b) , such that (f'(c))/(3c^(2))= (f(b)-f(a))/(b^(3)-a^(3)) .

Let f: [a, b] -> R be a function, continuous on [a, b] and twice differentiable on (a, b) . If, f(a) = f(b) and f'(a) = f'(b) , then consider the equation f''(x) - lambda (f'(x))^2 = 0 . For any real lambda the equation hasatleast M roots where 3M + 1 is

Let f:[a,b]rarr R be a function,continuous on [a,b] and twice differentiable on (a,b). If,f(a)=f(b) and f'(a)=f'(b), then consider the equation f''(x)-lambda(f'(x))^(2)=0. For any real lambda the equation hasatleast M roots where 3M+1 is