Prove that the curves `y=f(x),[f(x)>0],a n dy=f(x)sinx ,w h e r ef(x)` is differentiable function, have common tangents at common points.

Prove that the function f given by f(x)= |x-1|, x in R is not differentiable at x= 1.

Let g(x)=f(x)sinx ,w h e r ef(x) is a twice differentiable function on (-oo,oo) such that f(-pi)=1. The value of |g^n (-pi)| equals __________

y=f((2x-1)/(x^2+1)) and f^'(x)=sinx^2 then (dy)/(dx)

Prove that f(x)gi v e nb yf(x+y)=f(x)+f(y)AAx in R is an odd function.

If f(x)=x/(sinx)a n dg(x)=x/(tanx),w h e r e0ltxlt=1, then in this interval

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.

A function f: RvecR satisfies the equation f(x+y)=f(x)f(y) for all x , y in Ra n df(x)!=0fora l lx in Rdot If f(x) is differentiable at x=0a n df^(prime)(0)=2, then prove that f^(prime)(x)=2f(x)dot

Let F(x)=f(x)g(x)h(x) for all real x ,w h e r ef(x),g(x),a n dh(x) are differentiable functions. At some point x_0,F^(prime)(x_0)=21 F(x_0),f^(prime)(x_0)=4f(x_0),g^(prime)(x_0)=-7g(x_0), and h^(prime)(x_0)=kh(x_0) . Then k is________

Find the range of f(x)=(x-[x])/(1-[x]+x '),w h e r e[] represents the greatest integer function.

Find the domain and range of f(x)=sin^(-1)[x]w h e r[] represents the greatest function).