If `(x)=tan x`, the prove that `:f(x)+f(-x)=0`

If f (x) = cot x, then prove that : f(-x)=-f(x)

If f(x)=x|x|, then prove that f'(x)=2|x|

A function f:R rarr R satisfies the equation f(x+y)=f(x)f(y) for allx,y in R and f(x)!=0 for all x in R .If f(x) is differentiable at x=0 .If f(x)=2, then prove that f'(x)=2f(x) .

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

If f'(x)>0,f(x)>0AA x in(0,1) and f(0)=0,f(1)=1, then prove that 'f(x)f^(^^)-1(x)

If f(x)=|x|, prove that lim_(x rarr0)f(x)=0

If f(x+2a)=f(x-2a), then prove that f(x) is periodic.

If f: R to [0,∞) be a function such that f(x -1) + f(x + 1) = sqrt3(f(x)) then prove that f(x + 12) = f(x) .

(i) If f(x) = x + (1)/(x) prove that: [f(x)]^(3) = f(x^(3)) + 3f ((1)/(x)) (ii) If f(x) = x^(3) - (1)/(x^(3)) , prove that f(x) + f((1)/(x)) = 0 (iii) If f(x) = (1-x^(2))/(1+ x^(2)) , prove that f (tan theta) = cos 2 theta

Let f(x+y)=f(x)+f(y)+2xy-1 for all real x and f(x) be a differentiable function.If f'(0)=cosalpha, the prove that f(x)>0AA x in R