If x and y are positive reals then prove that `x < y hArr x^2 < y^2`.

If x,y,z are positive real numbers, prove that: sqrt(x^-1 y).sqrt(y^-1z).sqrt(z^(-1)x)=1

Let f(x) be periodic and k be a positive real number such that f(x+k)+f(x)=0fora l lx in Rdot Prove that f(x) is periodic with period 2kdot

If x=sectheta-tanthetaa n d y=cos e ctheta+cottheta, then prove that x y+1=y-x

If x,y and z are three real numbers such that x+y+z=4 and x^2+y^2+z^2=6 ,then show that each of x,y and z lie in the closed interval [2/3,2]

Prove that the straight lines x/a-y/b =m and x/a+y/b=1/m, where a and b are given positive real numbers and 'm' is a parameter, always meet on a hyperbola.

If x and y are real, show that the equation : sec^2 theta= (4xy)/(x+y)^2 is valid only when x = y ne 0 .

If x is a positive real number different from 1 such that log_a x, log_b x, log_c x are in A.P then

If z= x + iy, x, y real, prove that : |x|+|y| le sqrt2 |z| .

If sqrt((x/y)) + sqrt((y/x)) = lambda . Where x, y > 0 and lambda is a positive constant greater than or equal to 2 , prove that dy/dx = y/x

If x,y and z are real such that x+y+z=4, x^(2)+y^(2)+z^(2)=6 , x belongs to