1 Find the equation of the regression line for the given points. Round any final values to the nearest hundredth, if necessary.

x <- c(5.6, 6.3, 7, 7.7, 8.4)
y <- c(8.8, 12.4, 14.8, 18.2, 20.8)
regression_model <- lm(y ~ x)
intercept <- coef(regression_model)[1]
slope <- coef(regression_model)[2]

cat("y =", round(intercept, 2), "+", round(slope, 2), "x\n")
## y = -14.8 + 4.26 x

##2. Find all local maxima, local minima, and saddle points for the function given below. Write your answer(s) in the form ( x, y, z ). Separate multiple points with a comma. f ( x, y ) = 24x  6xy 2  8y 3

f <- function(x, y) {
  return(24*x - 6*x*y^2 - 8*y^3)
}

df_dx <- function(x, y) {
  return(24 - 6*y^2)
}

df_dy <- function(x, y) {
  return(-12*x*y - 24*y^2)
}

critical_points <- optim(c(0, 0), fn = function(xy) -f(xy[1], xy[2]), 
                         gr = function(xy) -c(df_dx(xy[1], xy[2]), df_dy(xy[1], xy[2])))$par

cat("Critical Point(s):", critical_points, "\n")
## Critical Point(s): 2.684172e+55 -3.183805e+55

3A grocery store sells two brands of a product, the “house” brand and a “name” brand. The manager estimates that if she sells the “house” brand for x dollars and the “name” brand for y dollars, she will be able to sell 81  21x + 17y units of the “house” brand and 40 + 11x  23y units of the “name” brand. 3. Step 1. Find the revenue function R ( x, y ). Step 2. What is the revenue if she sells the “house” brand for $2.30 and the “name” brand for $4.10?

x <- 2.30
y <- 4.10

R <- function(x, y) {
  units_house_brand <- 81 - 21*x + 17*y
  units_name_brand <- 40 + 11*x - 23*y
  return(units_house_brand*x + units_name_brand*y)
}

revenue <- R(x, y)

cat("Revenue $", x, "name brand for $", y, "is $", revenue)
## Revenue $ 2.3 name brand for $ 4.1 is $ 116.62

##4 A company has a plant in Los Angeles and a plant in Denver. The firm is committed to produce a total of 96 units of a product each week. The total weekly cost is given by C(x, y) = 1 6 x 2 + 1 6 y 2 + 7x + 25y + 700, where x is the number of units produced in Los Angeles and y is the number of units produced in Denver. How many units should be produced in each plant to minimize the total weekly cost?

library(optimx)
## Warning: package 'optimx' was built under R version 4.3.2
C <- function(x) {
  x_LA <- x[1]
  x_Denver <- x[2]
  return((1/6)*x_LA^2 + (1/6)*x_Denver^2 + 7*x_LA + 25*x_Denver + 700)
}
result <- optim(c(0, 0), C, control=list(fnscale=1))
x_LA_optimal <- result$par[1]
x_Denver_optimal <- result$par[2]

cat("Optimal units produced in Los Angeles:", x_LA_optimal, "\n")
## Optimal units produced in Los Angeles: -20.99461
cat("Optimal units produced in Denver:", x_Denver_optimal, "\n")
## Optimal units produced in Denver: -75.00166

5Evaluate the double integral on the given region.e 8x + 3y dA ; R: 2 £ x £ 4 and 2 £ y £ 4 Write your answer in exact form without decimals.

# Load necessary library
library(pracma)
## Warning: package 'pracma' was built under R version 4.3.1
f <- function(x, y) {
  return(exp(8*x + 3*y))
}
x_range <- c(2, 4)
y_range <- c(2, 4)
result <- integral2(f, x_range[1], x_range[2], y_range[1], y_range[2])
result
## $Q
## [1] 5.341559e+17
## 
## $error
## [1] 15214781905