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.