Calculus

Given the utility function of good x and y as, `\U=\x^{3/4}\y^{1/4}`

Prices of good X and Y `\Px=6\and\Py=3`

Budget `\B=$120`

The point of tangency or the combination of good X and Y (maximum satisfaction) `\U=\frac{dUx}{dUy}=frac{Px}{Py}`.....(1)

`\frac{dUx}{dx}=\x^{3/4}\y^{1/4}`

differentiating with respect to x,

`\=\3/4\x^{3/4-1}\y^{1/4}`

`\=\3/4\x^{-1/4}\y^{1/4}`......(ii)

`\frac{dUy}{dy}=\x^{3/4}\y^{1/4}`

differentiating with respect to y

`\=\x^{3/4}\1/4\y^{1/4-1}`

`\=\x^{3/4}\1/4\y^{-3/4}`.......(iii)

substituting the values in equation (i)

`\frac{3/4\y^{3/4}\y^{1/4}}{x^{3/4}\1/4\x^{-1/4}}=frac{6}{3}` 

Diving the upper portion of the equation by 3, we get;

 `\frac{1/4\y^{3/4}\y^{1/4}}{1/4\x^{3/4}\x^{1/4}}=frac{2}{3}` 

Cross multiplying;

`\3y=2x`

or, `\x=\frac{3y}{2}`.......(iv)

The equation of budget line can be written as `Px.X+Py.Y=120`........(v)

Substitution the equation (iv) prices of x and y good in equation (v), we get;

`\6\frac{3y}{2}+3y=120`

or, `\9y+3y=120`

or, `12y=120`

or, `y=10`

Similarly again substituting the value of y in equation (iv) weget;

`\x=\frac{3×10}{2}`

or, `\x=\frac{30}{2}`

or, `\x=15`

Hence the combination of 15 units of good x and 10 units of good y will be the point of tangency to the indifference curve.