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Q 4E

Expert-verifiedFound in: Page 238

Book edition
2nd

Author(s)
James Stewart

Pages
830 pages

ISBN
9781133112280

**The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares?**

The smallest possible value of the sum of the squares of the two positive numbers is 128.

1)The sum of two positive numbers is 16.

2)The sum of their squares is the smallest.

Let the numbers be *x* and *y*.

It is given that the sum of the two numbers is 16.

Therefore, it can be written as:

\(\begin{aligned}{c}x + y &= 16\\y &= 16 - x\end{aligned}\)

The function which represents the sum of the squares of two numbers is:

\(f\left( {x,y} \right) = {x^2} + {y^2}\) ……………(1)

Substitute \(y = 16 - x\)in equation (1)

\(f\left( x \right) = {x^2} + {\left( {16 - x} \right)^2}\)

\(f\left( x \right) = 2{x^2} - 32x + 256\)

A minimum of \(f\left( x \right)\) is obtained at \(f'\left( x \right) = 0\)

\(f\left( x \right) = 2{x^2} - 32x + 256\)

\(f'\left( x \right) = 4x - 32\)

Substitute \(f'\left( x \right) = 0\) as:

\(\begin{aligned}{c}4x - 32 &= 0\\4x &= 32\\x &= 8\end{aligned}\)

Therefore,

\(\begin{aligned}{c}y &= 16 - x\\y &= 16 - 8\\ &= 8\end{aligned}\)

And the minimum possible value the sum of the squares is:

\(\begin{aligned}{c}{8^2} + {8^2} &= 64 + 64\\ &= 128\end{aligned}\)

Thus, the smallest possible value of the sum of the squares of the two positive numbers is 128.

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