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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.
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
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