If `f(x-y),f(x)f(y),a n df(x+y)` are in A.P. for all `x , y ,a n df(0)!=0,` then (a)`f(4)=f(-4)` (b)`f(2)+f(-2)=0` (c)`f^(prime)(4)+f^(prime)(-4)=0` (d)`f^(prime)(2)=f^(prime)(-2)`

If f (x/y)= f(x)/f(y) , AA y, f (y)!=0 and f' (1) = 2 , find f(x) .

Let y=f(x) be a parabola, having its axis parallel to the y-axis, which is touched by the line y=x at x=1. Then, (a) 2f(0)=1-f^(prime)(0) (b) f(0)+f^(prime)(0)+f^(0)=1 (c) f^(prime)(1)=1 (d) f^(prime)(0)=f^(prime)(1)

A function f: Rvec[1,oo) satisfies the equation f(x y)=f(x)f(y)-f(x)-f(y)+2. If differentiable on R-{0}a n df(2)=5,f^(prime)(x)=(f(x)-1)/xdotlambdat h e nlambda= 2^(prime)f(1) b. 3f^(prime)(1) c. 1/2f^(prime)(1) d. f^(prime)(1)

If f(x+y)=f(x)dotf(y) for all real x , ya n df(0)!=0, then prove that the function g(x)=(f(x))/(1+{f(x)}^2) is an even function.

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

If f((x+y)/3)=(2+f(x)+f(y))/3 for all x,y f'(2)=2 then find f(x)

Let g(x) be the inverse of an invertible function f(x), which is differentiable for all real xdot Then g^('')(f(x)) equals. (a) -(f^('')(x))/((f^'(x))^3) (b) (f^(prime)(x)f^('')(x)-(f^(prime)(x))^3)/(f^(prime)(x)) (c) (f^(prime)(x)f^('')(x)-(f^(prime)(x))^2)/((f^(prime)(x))^2) (d) none of these

f(x+y)=f(x).f(y) for all x,yinR and f(5)=2,f'(0)=3 then f'(5) is equal to

If f(x)a n dg(x) are differentiable functions for 0lt=xlt=1 such that f(0)=10 ,g(0)=2,f(1)=2,g(1)=4, then in the interval (0,1)dot (a) f^(prime)(x)=0fora l lx (b) f^(prime)(x)+4g^(prime)(x)=0 for at least one x (c) f(x)=2g'(x) for at most one x (d)none of these

If f(x+f(y))=f(x)+yAAx ,y in Ra n df(0)=1, then find the value of f(7)dot