Question

Find the least value of 3x + 4y if x2y3 = 6.?

Find the least value of 3x + 4y if x2y3 = 6.?

Answers

10.

#1

f(x, y)= 3x + 4y and x2y3= 6

(x2)*(y3) = 6

=> x = √(6/y^3) = √6 /y^1.5

x = √6 * y^(- 1.5)

=> substitute in f(x, y)

=>f(y) = 3 * √6 * y^(- 1.5) + 4y

f'(y) = 3* √6 * (- 1.5) * y^(- 2.5) + 4

= - 4.5 * √6 * y^(- 2.5) + 4

At minimum>> f'(y) = 0

=> - 4.5 * √6 * y^(- 2.5) + 4 = 0

- 4.5 * √6 * y^(- 2.5) = - 4

square both sides

4.52 * 6 * y^(- 5) = 16

121.5 = 16 * y^5

y^5 = 121.5/16 = 7.59375

y = (7.59375)^(1/5) = 1.5

y = 1.5

=> x = √[6/(1.53)]

x = √(16/9)

x = 4/3

3x + 4y = 3*(4/3) + 4*(3/2) = 10

minumum of 10

#2