Question

One non-negative number is the reciprocal of another number. The sum of the two numbers is a minimum. What are the two numbers

Answer 1

The two numbers are 1 and 1.

Let the non-negative number be x and its reciprocal be y = 1/x

The sum of the two numbers is f(x,y) = x + y

= x + 1/x

= f(x)

For f(x) to be minimum, we differentiate it with respect to x and equate it to zero to find the value of x at which irt is minimum.

So, df(x)/dx = d(x + 1/x)/dx

= dx/dx + d(1/x)/dx

= 1 - 1/x²

Equating it to zero, we have

1 - 1/x² = 0

1 = 1/x²

x² = 1

Taking square-root of both sides, we have

x = ±√1

x = ±1

Since x is non-negative x = + 1

We differentiate f(x) again to determine if this value gives a minimum for f(x).

So, d²f(x)/dx² = d(1 - 1/x²)/dx

= d1/dx - d(1/x²)/dx

= 0 - × (-2/x³)

= 2/x³

Substituting x = 1 into the equation,

d²f(x)/dx² = 2/x³

= 2/1³

= 2 > 0.

So x = 1 is a minimum point for f(x)

Since x = 1 and y = 1/x, y = 1/1 = 1

So, the two numbers are 1 and 1.

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