Question

asked 2021-02-19

Multivariable optimization question. Find three positive real numbers whose sum is one and the sum of their squares is a minimum.

asked 2020-12-17

A surface is represented by the following multivariable function,

\(\displaystyle{f{{\left({x},{y}\right)}}}=\frac{{1}}{{3}}{x}^{{3}}+{y}^{{2}}-{2}{x}{y}-{6}{x}-{3}{y}+{4}\)

a) Calculate \(\displaystyle{f}_{{\times}},{f}_{{{y}{x}}},{f}_{{{x}{y}}}{\quad\text{and}\quad}{f}_{{{y}{y}}}\)

b) Calculate coordinates of stationary points.

c) Classify all stationary points.

\(\displaystyle{f{{\left({x},{y}\right)}}}=\frac{{1}}{{3}}{x}^{{3}}+{y}^{{2}}-{2}{x}{y}-{6}{x}-{3}{y}+{4}\)

a) Calculate \(\displaystyle{f}_{{\times}},{f}_{{{y}{x}}},{f}_{{{x}{y}}}{\quad\text{and}\quad}{f}_{{{y}{y}}}\)

b) Calculate coordinates of stationary points.

c) Classify all stationary points.

asked 2021-02-18

A surface is represented by the following multivariable function,

\(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{3}}+{y}^{{3}}-{3}{x}-{3}{y}+{1}\)

Calculate coordinates of stationary points.

\(\displaystyle{f{{\left({x},{y}\right)}}}={x}^{{3}}+{y}^{{3}}-{3}{x}-{3}{y}+{1}\)

Calculate coordinates of stationary points.

asked 2020-11-27

Comsider this, multivariable fucntion

\(\displaystyle{f{{\left({x},{y}\right)}}}={3}{x}{y}-{x}^{{2}}+{5}{y}^{{2}}-{25}\)

a)What is the value of f(-1,3)?

b)Find all x-values such that f(x,x)=0

\(\displaystyle{f{{\left({x},{y}\right)}}}={3}{x}{y}-{x}^{{2}}+{5}{y}^{{2}}-{25}\)

a)What is the value of f(-1,3)?

b)Find all x-values such that f(x,x)=0

asked 2021-02-09

Consider this multivariable function. f(x,y)=xy+2x+y−36

a) What is the value of f(2,−3)?

b) Find all x-values such that f (x,x) = 0

a) What is the value of f(2,−3)?

b) Find all x-values such that f (x,x) = 0

asked 2020-11-20

Consider this multivariable function.
\(\displaystyle{f{{\left({x},{y}\right)}}}={y}{e}^{{{3}{x}}}+{y}^{{2}}\)

a) Find \(\displaystyle{{f}_{{y}}{\left({x},{y}\right)}}\)

b) What is value of \(\displaystyle{{f}_{{\times}}{\left({0},{3}\right)}}\)?

a) Find \(\displaystyle{{f}_{{y}}{\left({x},{y}\right)}}\)

b) What is value of \(\displaystyle{{f}_{{\times}}{\left({0},{3}\right)}}\)?

asked 2021-01-05

A concert promoter produces two kinds of souvenir shirt, one kind sells for $18 ad the other for $25. The company determines, the total cost, in thousands of dollars, of producting x thousand of the $18 shirt and y thousand of the $25 shirt is given by

\(\displaystyle{C}{\left({x},{y}\right)}={4}{x}^{{2}}-{6}{x}{y}+{3}{y}^{{2}}+{20}{x}+{19}{y}-{12}.\)

How many of each type of shirt must be produced and sold in order to maximize profit?

\(\displaystyle{C}{\left({x},{y}\right)}={4}{x}^{{2}}-{6}{x}{y}+{3}{y}^{{2}}+{20}{x}+{19}{y}-{12}.\)

How many of each type of shirt must be produced and sold in order to maximize profit?

asked 2021-01-06

Let \(\displaystyle{z}{\left({x},{y}\right)}={e}^{{{3}{x}{y}}},{x}{\left({p},{q}\right)}=\frac{{p}}{{q}}{\quad\text{and}\quad}{y}{\left({p},{q}\right)}=\frac{{q}}{{p}}\) are functions. Use multivariable chain rule of partial derivatives to find

(i) \(\displaystyle\frac{{\partial{z}}}{{\partial{p}}}\)

(ii) \(\displaystyle\frac{{\partial{z}}}{{\partial{q}}}\).

(i) \(\displaystyle\frac{{\partial{z}}}{{\partial{p}}}\)

(ii) \(\displaystyle\frac{{\partial{z}}}{{\partial{q}}}\).

asked 2021-01-13

Average value over a multivariable function using triple integrals. Find the average value of \(\displaystyle{F}{\left({x},{y},{z}\right)}={x}^{{2}}+{y}^{{2}}+{z}^{{2}}\) over the cube in the first octant bounded bt the coordinate planes and the planes x=5, y=5, and z=5

asked 2021-01-05

Let \(\displaystyle{z}={e}^{{{x}^{{2}}+{3}{y}}}+{x}^{{3}}{y}^{{2}}\), where \(\displaystyle{x}={t}{\cos{{r}}}{\quad\text{and}\quad}{y}={r}{t}^{{4}}\). Use the chain rule fot multivariable functions (Cals III Chain Rule) to find \(\displaystyle\frac{{\partial{z}}}{{\partial{r}}}\) and \(\displaystyle\frac{{\partial{z}}}{\partial}{t}{)}\). Give your answers in terms of r and t only. Be sure to show all of your work.