Question

The sum of two positive numbers is 24. Find the two numbers if the sum of their squares is minimum.​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

SOLUTION

GIVEN

• The sum of two positive numbers is 24.

• The sum of their squares is minimum.

TO DETERMINE

The numbers

EVALUATION

Let the numbers are x and y

By the given condition 1

x + y = 24 - - - - - - - (1)

Let S = x² + y²

Then we have

$\displaystyle \sf{ S = {x}^{2} + {(24 - x)}^{2} }$

Differentiating both sides with respect to x we get

$\displaystyle \sf{ \frac{dS}{dx} = 2x - 2(24 - x) }$

$\displaystyle \sf{ \implies \frac{dS}{dx} = 4x -48}$

Again Differentiating both sides with respect to x

$\displaystyle \sf{ \frac{ {d}^{2} S}{d {x}^{2} } = 4}$

For extremum we have

$\displaystyle \sf{ \implies \frac{dS}{dx} = 0}$

$\displaystyle \sf{ \implies 4x -48 = 0}$

$\displaystyle \sf{ \implies 4x = 48}$

$\displaystyle \sf{ \implies x = 12}$

Now for x = 12 we have

$\displaystyle \sf{ \frac{ {d}^{2} S}{d {x}^{2} } > 0 }$

Thus for x = 12 , S has minimum value

When x = 12 we have y = 24 - 12 = 12

For x = y = 12 we have

S = x² + y² = 12² + 12² = 144 + 144 = 288

FINAL ANSWER

Hence the required numbers are 12 , 12

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