Statement 1: For `f(x)=sinx ,f^(prime)(pi)=f^(prime)(3pi)dot` Statement 2: For `f(x)=sinx ,f(pi)=f(3pi)dot`

Suppose the function f(x) satisfies the relation f(x+y^3)=f(x)+f(y^3)dotAAx ,y in R and is differentiable for all xdot Statement 1: If f^(prime)(2)=a ,t h e nf^(prime)(-2)=a Statement 2: f(x) is an odd function.

Statement 1: If f(x) is an odd function, then f^(prime)(x) is an even function. Statement 2: If f^(prime)(x) is an even function, then f(x) is an odd function.

If f(x)=|x|^(|sinx|), then find f^(prime)(-pi/4)

f(x)=sinx+int_(-pi//2)^(pi//2)(sinx+tcosx)f(t)dt The range of f(x) is

A nonzero polynomial with real coefficient has the property that f(x)=f^(prime)(x)dot f^(prime prime)(x)dot If a is the leading coefficient of f(x), then the value of 1/(2a) is____

Let f(x0 be a non-constant thrice differentiable function defined on (-oo,oo) such that f(x)=f(6-x)a n df^(prime)(0)=0=f^(prime)(x)^2=f(5)dot If n is the minimum number of roots of (f^(prime)(x)^2+f^(prime)(x)f^(x)=0 in the interval [0,6], then the value of n/2 is___

Statement 1: Let f(x)=sin(cos x)in[0,pi/2]dot Then f(x) is decreasing in [0,pi/2] Statement 2: cosx is a decreasing function AAx in [0,pi/2]

Statement 1: If f(0)=0,f^(prime)(x)=ln(x+sqrt(1+x^2)), then f(x) is positive for all x in R_0dot Statement 2: f(x) is increasing for x >0 and decreasing for x<0

If fx=x+sinx , then int_(pi)^(2pi)f^(-1)(x)dx is equal to