AEB $3510$: Problem Set #$8-$ Optimization of multivariate functions

$1$

$ma{x}_{x_1,x_2}f(x_1,x_2)=160x_1-3x_1^2-2x_1{x}_2-2x_2^2+120x_2-18$

$($a$)$ Find the stationary point$($s$)$ that satisfy the first$-$order necessary conditions for a relative maximizer.

$($b$)$ Using the second$-$order partial derivatives, confirm that the stationary point$($s$)$ is $($are$)$ a relative maximizer.

$ma{x}_{x_1,x_2}f(x_1,x_2)=42x_1-4x_1^2-2x_1{x}_2+18x_2-\frac{3}{2}{x}_2^2$

$($a$)$ Find the stationary point$($s$)$ that satisfy the first$-$order necessary conditions for a relative maximizer.

$($b$)$ Take the second$-$order $($direct$)$ partial derivatives to confirm that the stationary point$($s$)$ is $($are$)$ a relative maximizer.

Suppose you are a rational, utility maximizing consumer trying to determine how to spend your money between purchasing Taylor Swift songs off iTunes $(t)$ and other goods $(x).$ You have a Cobb$-$Douglas utility function given by $u(t,x)=t^{0\mathrm{.}15}{x}^{0\mathrm{.}85}.$ You have a fixed amount of money $($$$2,000)$ to spend on these two goods. Suppose the Taylor Swift songs cost $$1\mathrm{.}50$ on iTunes, and the price of all other goods are normalized at $$1.$