If f"(x) exists for all points in `[a,b]` and
`(f(c )-f(a))/(c-a)=(f(b)-f( c))/(b-c),"where"a lt clt b,` then show that there exists a number 'k' such that f"(k)=0.

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

Let f be a continuous function on [a , b]dot If F(x)=(int_a^xf(t)dt-int_x^bf(t)dt)(2x-(a+b)), then prove that there exist some c in (a , b) such that int_a^cf(t)dt-int_c^bf(t)dt=f(c)(a+b-2c)dot

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)a n dg(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. |f(a)f(b)g(a)g(b)|=(b-a)|f(a)f^(prime)(c)g(a)g^(prime)(c)|,w h e r ea

If f(x) is differentiate in [a,b], then prove that there exists at least one c in (a,b)"such that"(a^(2)-b^(2))f'(c)=2c(f(a)-f(b)).

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

Let f be continuous on [a , b],a >0,a n d differentiable on (a , b)dot Prove that there exists c in (a , b) such that (bf(a)-af(b))/(b-a)=f(c)-cf^(prime)(c)

Let f be a real-valued function defined on interval (0,oo) ,by f(x)=lnx+int_0^xsqrt(1+sint).dt . Then which of the following statement(s) is (are) true? (A). f"(x) exists for all in (0,oo) . " " (B). f'(x) exists for all x in (0,oo) and f' is continuous on (0,oo) , but not differentiable on (0,oo) . " " (C). there exists alpha>1 such that |f'(x)|<|f(x)| for all x in (alpha,oo) . " " (D). there exists beta>1 such that |f(x)|+|f'(x)|<=beta for all x in (0,oo) .

Let f: RvecR ,f(x)=(x-a)/((x-b)(x-c)),b > cdot If f is onto, then prove that a in (b , c)dot