" UPU Prove the inequalities "(n!)^(2)<=n^(n)(n!)<(2n)!" for all positive integers "n" ."

Prove the inequalities x-x^(2) lt In (1+x) "at" x gt 0

Prove inequalities using LMVT

If s be the sum of n positive unequal quantities a, b, c then prove the inequality , (s)/(s-a)+(s)/(s-b)+(s)/(s-c)+…… gt (n^2)/(n-1) (n ge 2) .

By mathematical induction prove the inequality (1+x)^(n)ge1+nx for all ninNN and xgt-1 .

Use calculus to prove the inequality,sin x>=(2x)/(pi) in 0<=x<=(pi)/(2)

Prove the inequalities tan x gt x + x^(3) //3 " at" 0 lt x lt pi//2

Let a gt 1 and n in N . Prove that inequality , a^(n)-1 ge n (a^((n+1)/(2))-a^((n-1)/(2))) .