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).

f is continuous in [a, b] and differentiable in (a, b) (where a>0 ) such that f(a)/a=f(b)/b. Prove that there exist x_0 in (a, b) such that f'(x_0 ) = f(x_0)/x_0

Statement 1: If 27 a+9b+3c+d=0, then the equation f(x)=4a x^3+3b x^2+2c x+d=0 has at least one real root lying between (0,3)dot Statement 2: If f(x) is continuous in [a,b], derivable in (a , b) such that f(a)=f(b), then there exists at least one point c in (a , b) such that f^(prime)(c)=0.

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]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

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'(c))/(3c^(2))=(f(b)-f(a))/(b^(3)-a^(3))

Let f be continuous on [a;b] and differentiable on (a;b) If f(x) is strictly increasing on (a;b) then f'(x)>0 for all x in(a;b) and if f(x) is strictly decreasing on (a;b) then f'(x)<0 for all x in(a;b)