Find out the answer to "Among all pairs of numbers (x,y) such that..."

rastafarral6

Answered question

2021-11-15

Among all pairs of numbers (x,y) such that

4 x + 2 y = 18

find the pair for which the sum of squares,

x^{2} + y^{2}

, is minimum. Write your answers as fractions reduced to lowest terms.

juniorekze

Beginner2021-11-16Added 18 answers

Step 1
Given

4 x + 2 y = 18

y = \frac{18 - 4 x}{2} = 9 - 2 x

f = x^{2} + y^{2}

Put the value of y in f

f = x^{2} + {(9 - 2 x)}^{2}

f = x^{2} + 81 + 4 x^{2} - 36 x

= 5 x^{2} - 36 x + 81

Step 2
Differentiate with respect to x

f^{'} = 10 x - 36

Find critical points

f^{'} = 0

10 x - 36 = 0

x = \frac{18}{5}

f 10

Step 3
Since f'' is positive. Therefore minimum at

x = \frac{18}{5}

y = 9 - 2 \times \frac{18}{5}

= \frac{45 - 36}{5} = \frac{9}{5}

Minimum Pair

(\frac{18}{5}, \frac{9}{5})

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