If `'f'` is an increasing function from `RvecR` such that `f^(x)>0a n df^(-1)` exists then `(d^2(f^(-1)(x)))/(dx^2)` is `<0` b. `>0` c. `=0` d. cannot be determined

If f(x) is monotonically increasing function for all x in R, such that f''(x)gt0andf^(-1)(x) exists, then prove that (f^(-1)(x_(1))+f^(-1)(x_(2))+f^(-1)(x_(3)))/(3)lt((f^(-1)(x_(1)+x_(2)+x_3))/(3))

If f(x)+f(2-x)=0 , then int_(0)^(2)(1)/(1+2^(f(x)))dx=

If f(x)+f(2-x)=0 , then int_(0)^(2)(1)/(1+2^(f(x)))dx=

If f: RvecR is a differentiable function such that f^(prime)(x)>2f(x)fora l lxRa n df(0)=1,t h e n : f(x) is decreasing in (0,oo) f^(prime)(x) e^(2x)in(0,oo)

If f(x) is a differentiable real valued function such that f(0)=0 and f\'(x)+2f(x) le 1 , then (A) f(x) gt 1/2 (B) f(x) ge 0 (C) f(x) le 1/2 (D) none of these

If f: RvecR is a twice differentiable function such that f"(x)>0fora l lxR ,a n d f(1/2)=1/2,f(1)=1 then: f^(prime)(1)>1 (b) f^(prime)(1)lt=0 (c)1/2 ltf^(prime)(1)lt1 (d)="" 0ltf^(prime)(1)lt="1/2

Let f(x) be a continuous function in R such that f(x)+f(y)=f(x+y) , then int_-2^2 f(x)dx= (A) 2int_0^2 f(x)dx (B) 0 (C) 2f(2) (D) none of these

Consider f(x) is a real valued function f(x) such that f(x+y)=f(x)+f(y)+2xy, such that f(x) is differentiable for all xinR and (df)/(dx)(0)=0,then answer the following

If f'(0)=0 and f(x) is a differentiable and increasing function,then lim_(x rarr0)(x*f'(x^(2)))/(f'(x))

If f(x) is differentiable and strictly increasing function,then the value of lim_(x rarr0)(f(x^(2))-f(x))/(f(x)-f(0)) is 1 (b) 0(c)-1 (d) 2