Let `f and g` be function continuous in `[a,b] `and differentiable on `[a,b]`.If `f(a)=f(b)=0` then show that there is a point `c in (a,b)` such that `g'(c) f(c)+f'(c)=0`.

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))

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)) .

PARAGRAPH : If function f,g are continuous in a closed interval [a,b] and differentiable in the open interval (a,b) then there exists a number c in (a,b) such that [g(b)-g(a)]f'(c)=[f(b)-f(a)]g'(c) If f(x)=e^(x) and g(x)=e^(-x) , a<=x<=b c=

Let f be continuous on [a,b],a>0, and differentiable on (a,b). Prove that there exists c in(a,b) such that (bf(a,b))/(b-a)=f(c)-cf'(c)

Let f and g be differentiable on [0,1] such that f(0)=2,g(0),f(1)=6 and g(1)=2. Show that there exists c in(0,1) such that f'(c)=2g'(c)

Let f(x), be a function which is continuous in closed interval [0,1] and differentiable in open interval 10,1[ Show that EE a point c in]0,1[ such that f'(c)=f(1)-f(0)

If f is continuous on [a,b] and differentiable in (a,b) then there exists c in(a,b) such that (f(b)-f(a))/((1)/(b)-(1)/(a)) is

If f(x) and g(x) are continuous functions in [a,b] and are differentiable in (a,b) then prove that there exists at least one c in(a,b) for which.det[[f(a),f(b)g(a),g(b)]]=(b-a)det[[f(a),g'(c)]], where a