Answer :

Given,


The two numbers are positive.


the product of two numbers is 49.


the sum of the two numbers is minimum.


Let us consider,


x and y are the two numbers, such that x > 0 and y > 0


Product of the numbers : x × y = 49


Sum of the numbers : S = x + y


Now as,


x × y = 49


------ (1)


Consider,


S = x + y


By substituting (1), we have


------ (2)


For finding the maximum/ minimum of given function, we can find it by differentiating it with x and then equating it to zero. This is because if the function f(x) has a maximum/minimum at a point c then f’(c) = 0.


Differentiating the equation (2) with x




----- (3)


[Since and ]


Now equating the first derivative to zero will give the critical point c.


So,





= x2 = 49



As x > 0, then x = 7


Now, for checking if the value of S is maximum or minimum at x=7, we will perform the second differentiation and check the value of at the critical value x = 7.


Performing the second differentiation on the equation (3) with respect to x.





[Since and ]



Now when x = 7,



As second differential is positive, hence the critical point x = 7 will be the minimum point of the function S.


Therefore, the function S = sum of the two numbers is minimum at x = 7.


From Equation (1), if x= 7



Therefore, x = 7 and y = 7 are the two positive numbers whose product is 49 and the sum is minimum.


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