• 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
x × y = 49
S = x + y
By substituting (1), we have
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
[Since and ]
Now equating the first derivative to zero will give the critical point c.
= 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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