If f(x) = `ax^(3),` show that xf'(x) - 3f(x) = 0.

If f(x)=x^3-1/(x^3) , show that f(x)+f(1/x)=0.

If f(x)=|x| then show that f'(3) =1

If f(x)=x+(1)/(x) , then prove that : {f(x)}^(3)=f(x^(3))+3*f((1)/(x))

A function f : R to R is defined by f(x) = 2x^(3) + 3 . Show that f is a bijective mapping .

Let f(x) be a function defined by f(x) = {{:((3x)/(|x|+2x),x ne 0),(0,x = 0):} Show that lim_(x rarr 0) f(x) does not exist.

Let f(x),xgeq0, be a non-negative continuous function, and let f(x)=int_0^xf(t)dt ,xgeq0, if for some c >0,f(x)lt=cF(x) for all xgeq0, then show that f(x)=0 for all xgeq0.

"If " (d)/(dx)f(x)=f'(x), " then " int(xf'(x)-2f(x))/(sqrt(x^(4)f(x)))dx is equal to

Consider f : R->{-4/3}->R-{4/3} given by f(x) = (4x+3)/(3x+4) . Show that f is bijective. Find the inverse of f and hence find f^(-1)(0) and x such that f^-1(x) = 2 .

If for all x,y the function f is defined by f(x)+f(y)+f(x).f(y)=andf(x)gt0. Then show f'(x)=0 when f'(x)=0 is differentiable.

if f(x)=3e^(x^2) then f'(x)-2xf(x)+1/3f(0)-f'(0)