let `f(x)=2+cosx` for all real x. Statement 1: For each real t, there exists a point c in `[t,t+pi]` such that `f '(c)=0`. Statement 2: `f(t)=f(t+2pi)` for each real t

Statement 1: If both functions f(t)a n dg(t) are continuous on the closed interval [a,b], differentiable on the open interval (a,b) and g^(prime)(t) is not zero on that open interval, then there exists some c in (a , b) such that (f^(prime)(c))/(g^(prime)(c))=(f(b)-f(a))/(g(b)-g(a)) Statement 2: If f(t)a n dg(t) are continuous and differentiable in [a, b], then there exists some c in (a,b) such that f^(prime)(c)=(f(b)-f(a))/(b-a)a n dg^(prime)(c)(g(b)-g(a))/(b-a) from Lagranges mean value theorem.

Let f(x)=(1-x)^(2) sin^(2) x+x^(2) for all x in RR , and let g(x)=int_(1)^(x) ((2(t-1))/(t+1)-log t)f(t) dt for all x in (1, oo) . Consider the statements : P : there exists some x in RR such that f(x)+2x=2(1+x^(2)) Q : There exists some x in RR such that 2f(x)+1=2x(1+x) then-

Let f((x+y)/2)=(f(x)+f(y))/2 for all real x and y. If f'(0) exists and equals-1 and f(0)=1, find f(2)

Let f((x+y)/(2))=1/2 |f(x) +f(y)| for all real x and y, if f '(0) exists and equal to (-1), and f(0)=1 then f(2) is equal to-

Let (f(x+y)-f(x))/2=(f(y)-a)/2+x y for all real xa n dydot If f(x) is differentiable and f^(prime)(0) exists for all real permissible value of a and is equal to sqrt(5a-1-a^2)dot Then (a) f(x) is positive for all real x (b) f(x) is negative for all real x (c) f(x)=0 has real roots (d)Nothing can be said about the sign of f(x)

Let f((x+y)/(2))=(1)/(2)[f(x)+f(y)] for all real x and y. If f'(0) exists and equals (-1),f(0)=1 , find f(2).

Let f:R to R be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt . f(x) increases for

Let f:RtoR be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt . y=f(x) is

Let F(x) be an indefinite integral of sin^(2) x Statement - I : The function F(x) satisfies F(x+pi) = F(x) for all real x Statement - II : sin^(2) (pi+x) = sin^(2) x for all real x

If at x = t, the derivative of f(x) is f’(t) then.the value of lim_(xtot) (xf(t)-tf(x))/(x-t) is