Prove that `[(a^2+b^2)/(a+b)]^(a+b)> a^a b^b >{(a+b)/2}^(a+b)dot`

Prove that [(a^(2) + b^(2))/(a + b)]^(a + b) gt a^(a) b^(b) gt {(a + b)/(2)}^(a + b) , where a,b gt 0

Prove that |[(b+c)^2, a^2, bc],[(c+a)^2, b^2, ca],[(a+b)^2, c^2, ab]|=(a-b)(b-c)(c-a)(a+b+c)(a^2+b^2+c^2)

If f(x)=((a+x)/(b+x))^(a+b+2x) then prove that f'(0)=[2log((a)/(b))+(b^(2)-a^(2))/(ab)]((a)/(b))^(a+b)

If a,b,c are in G.P.then prove that (a^(2)+ab+b^(2))/(bc+ca+ab)=(b+a)/(c+b)

If a,b,c are in GP, prove that (a^2-b^2)(b^2+c^2)=(b^2-c^2)(a^2+b^2) .

Prove that matrix [((b^(2)-a^(2))/(a^(2)+b^(2)),(-2ab)/(a^(2)+b^(2))),((-2ab)/(a^(2)+b^(2)),(a^(2)-b^(2))/(a^(2)+b^(2)))] is orthogonal.

Prove that |(1,a,a^2),(1,b,b^2),(1,c,c^2)|=(a-b)(b-c)(c-a)

Prove that : (i) |{:(a,c,a+c),(a+b,b,a),(b,b+c,c):}|=2 abc (ii) Prove that : |{:(a^(2),bc,ac+c^(2)),(a^(2)+ab,b^(2),ac),(ab,b^(2)+bc,c^(2)):}|=4a^(2)b^(2)c^(2)

If |vec a|=a and |vec b|=b then prove that ((vec a)/(a^(2))-(vec b)/(b^(2)))=((vec a-vec b)/(ab))^(2)