The two numbers A and B (with A≤B) whose difference is 36 and whose product is minimized are;
A = -18 and B = 18
Let the two numbers be A and B.
A ≤ B
Thus;
B - A = 36 ---(eq 1)
P = BA ----(eq 2)
Making B the subject in eq 1 gives;
B = 36 + A
put 36 + A for B in eq 2 to get;
P = A(36 + A)
P = A² + 36A
The values of A and B if their product is minimized is gotten by finding the derivative of the product and equating to zero to get;.
P' = 2A + 36
Equating P' to zero gives;
2A + 36 = 0
2A = -36
A = -36/2
A = -18
Thus;
B = 36 + A
B = 36 - 18
B = 18
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