a+b=1a and b are positive real nos.Prove that: (a^a)(b^b)+(a^b)(b^a)>1.

Dear Pratik,

Given that a + b = 1.

That is 

We need to prove that 

Now substitute  in the expression :

Thus,

Observe the above expression.

Since a and b are positivie real numbers less than 1, the product 

Now let us observe the term in the square bracket.

Since  is negative.

Thus, the power of the first term is negative and hence the first term, that is, 

Similaraly, the power of the second term is negative and hence the second term, that is 

Thus, the sum of two numbers which are less than 1 is also less than 1.

That is the term inside the square bracket, 

Thus, the product,

Thus, the expression,

and it is equal to 1 if 

Therefore, we have 

Cracking IIT just got more exciting,It s not just all about getting assistance from IITians, alongside Target Achievement and Rewards play an important role. ASKIITIANS has it all for you, wherein you get assistance only from IITians for your preparation and win by answering queries in the discussion forums. 

https://www.askiitians.com/packages/packages.aspx

So start the brain storming…. become a leader with Elite Expert League ASKIITIANS

Thanks

Aman Bansal

Askiitian Expert