Find two positive numbers whose product is 100 and whose sum is a minimum. • Identify the Objective Function (OF) ?(?,?) • Identify the Constraint • Use the Constraint to express the OF by a function ?(?) •Identify the Interval of Interest • Find the ?-value(s) of a Critical Point(s) • Apply the Second Derivative Test (SDT) • Apply the Extreme Value Theorem (EVT) • Show both numbers
Find two positive numbers whose product is 100 and whose sum is a minimum.
• Identify the Objective Function (OF) ?(?,?)
• Identify the Constraint
• Use the Constraint to express the OF by a function ?(?)
•Identify the Interval of Interest
• Find the ?-value(s) of a Critical Point(s)
• Apply the Second Derivative Test (SDT)
• Apply the Extreme Value Theorem (EVT)
• Show both numbers
Suppose x and y are the two positive numbers. Then, xy = 100 and their sum x+y is minimum.
Objective Function :
minimize (x+y)
Constraint :
xy = 100 , x>0, y>0
From the constraint we have, y= 100/x ; putting it into the objective function we get minimize (x+100/x) . So, OF can be written as a function of one variable as : minimize f(x) = x+100/x
Differentiating f(x) and equating it to zero, we will get the critical points.
Now, f'(x) = 0 gives (1-100/x^2) = 0 , which gives x^2 = 100, i.e. x = 10, -10. But x being a positive number we neglect the negative value of x. Hence, x = 10.
Clearly, interval of interest is any interval that contains the point x = 10 and excludes the point x=0, since f becomes very large near 0.
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