determine the amounts of food and clothing within the restricted budget that maximize the consumer's utility

Assume that the prices of goods X and Y are fixed at $ 15 and $ 10 respectively and that the representative consumer has a monthly fixed income of $ 5000. Based on this information, determine the amounts of food and clothing within his restricted budget that maximize the consumer’s utility

U (x,y) = 20. x3. y5

(M) = $ 5000

Px = $ 15

Py = $ 10

so:

MaxU (x,y) = U(x,y)

MaxU (x,y) = 20. x³. y⁵

Y= M – Px . x

Y = 5000 – 15 x

Y= 500 – 1,5 x

MaxU (x,y) = U(x,y)

MaxU (x,y) = 20x³ . (500 – 1,5x)⁵