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`

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)

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)

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.

Let f(x)be continuous on [a,b], differentiable in (a,b) and f(x)ne0"for all"x in[a,b]. Then prove that there exists one c in(a,b)"such that"(f'(c))/(f(c))=(1)/(a-c)+(1)/(b-c).

Let f(x) be a function such that its derovative f'(x) is continuous in [a, b] and differentiable in (a, b). Consider a function phi(x)=f(b)-f(x)-(b-x)f'(x)-(b-x)^(2) A. If Rolle's theorem is applicable to phi(x) on, [a,b], answer following questions. If there exists some unmber c(a lt c lt b) such that phi'(c)=0 and f(b)=f(a)+(b-a)f'(a)+lambda(b-a)^(2)f''(c) , then lambda is

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 a , b , c be three real numbers such that a < b < c . f(x) is continuous in [a , c] and differentiable in (a , c) Also, f^(prime)(x) is strictly increasing in (a , c) Prove that (b-c)f(a)+(c-a)f(b)+(a-b)f(c)<0.

Let a , b , c be three real numbers such that a

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) .