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

a+b=1


a and b are positive real nos.


Prove that: (a^a)(b^b)+(a^b)(b^a)>1.

Grade:12

1 Answers

Aman Bansal
592 Points
10 years ago

 

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 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 

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