Prove that `(ab+by)gt 4abxy,` where `a,b,x,ygt0`.

Prove that (a b+x y)(a x+b y)>4a b x y(a , b ,x ,y >0)dot

Using Lagranges mean value theorem, prove that (b-a)/bltlog(b/a)lt(b-a)/a="" a, where 0ltaltbdot

Prove that [lim_(xto0) (sinx)/(x)]=0, where [.] represents the greatest integer function.

Prove that (1)/(a) + (1)/(b) + (1)/(c ) ge (1)/(sqrt((bc))) + (1)/(sqrt((ca))) + (1)/(sqrt((ab))) , where a,b,c gt 0

The area enclosed by the curves ax +- by +- c =0 where a, b, c gt 0

For 0ltxlt(pi)/(2) ,prove that x gt sin x and hence cos (sin x) gt sin (cosx)

If a+b+c=0 then check the nature of roots of the equation 4a x^2+3b x+2c=0 where a ,b ,c in R

Prove that [lim_(xto0) (tan^(-1)x)/(x)]=0, where [.] represents the greatest integer function.