If x and y are positive quantities whose sum is 4 , show that `(x+1/x)^(2)+(y+1/y)^(2)ge 12 1/2 `

If x+y+z=1a n dx ,y ,z are positive, then show that (x+1/x)^2+(y+1/y)^2+(z+1/z)^2>(100)/3

If x and y are positive real numbers and m, n are any positive integers, then prove that (x^n y^m)/((1+x^(2n))(1+y^(2m))) lt 1/4

If x,y are positive real numbers and m, n are positive integers, then prove that (x^(n) y^(m))/((1 + x^(2n))(1 + y^(2m))) le (1)/(4)

If a, b, x, y are positive natural numbers such that (1)/(x) + (1)/(y) = 1 then prove that (a^(x))/(x) + (b^(y))/(y) ge ab .

If "sec"^(-1)x="cosec"^(-1)y , show that 1/(x^(2))+1/(y^(2))=1

If 2^x = 3^y =12^z show that 1/z= 1/y+2/x

If x=sint,y=sinKt then show that (1-x^(2))y_2-xy_(1)+K^(2)y=0 .

If x=sin(1/alogy) , show that (1-x^2)y_2-x\ y_1-a^2\ y=0 .

For all positive values of x and y, the value of ((1+x+x^(2))(1+y+y^(2)))/(xy) ,is

If y=sin^(-1)x , show that (d^2y)/(dx^2)=x/((1-x^2)^(3//2))