Given f(x) =x³ – 81x + 10, use rates of change to verify that there is an optimal value at x = -9. Mathematically demonstrate whether this optimal value is a maximum or a minimum and briefly explain your reasoning.

Q: The cost per unit of producing a product is 40+5 dollars, where x represents the number of units…

A: Consider the provided cost for 1 unit of the product, 40 + 5x dollars. For x units of the product,…

Q: A retailer has been selling 2,000 computers a week at $850 each. The marketing department predicts…

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Q: A coffee shop finds that at the price of $3.00, demand for their popular cappuccino drink is 3000…

A: Given that: At price of $3.00, demand = 3000 cups a year.For every $0.25 decrease in price, demand…

Q: A 440-unit hotel is trying to determine their daily rates. From experience, the hotel manager knows…

A: Let x = number of dollar increase in rent per day. Total room rented = (440 – x) rooms Rent per day…

Q: Given the piecewise function: 4x + 12 { x<-2 - 1<x f(x) = -2x + 6 18. What is the interval of the…

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Q: Suppose that the selling price p of an item for the quantity x sold is given by the function 1 x +…

A: Given  Demand equation of x  p = -x8+26    ......... (1) Revenue function  R(x) R= p.x Putting the…

Q: What production level will yield a maximum revenue?

A: R=−x^3+30x^2+900x   To maximize R, find dR/dx and set to 0 : dR/dx = -3x² + 60x + 900 0 = -3x² + 60x…

Q: What is the minimum point of the curve y = x2 – 6x + 4

A: The given function is: y=x2-6x+4   First, we have to find the critical points by solving dydx=0.

Q: A firm can produce 300 units per week. If its total cost function is C = 800 + 1100x dollars and its…

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Q: A manufacturer estimates that it's product can be produced at a cost of C(X)=40000-100x+x^3. If the…

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Q: If the total revenue function for a computer is R(x) = 1800x – 23x2 - x³, find the level of sales,…

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Q: A manufacturer has been selling gadgets at a price of $50 each, and at this price, customers have…

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Q: suppose a baby food company has determined that its total revenue R for its food is given by R=…

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Q: Given the piecewise function: 4x + 12 x<-2 - 1<x f(x) = -2x + 6 -2 What is the relative maximum?…

A: We first sketch the function

Q: A firm can produce 100 units per week. If its total cost function is C = 600 + 1200x dollars and its…

A: Profit = Revenue - Cost  P(x)=(1300x-x^2)-(600+1200x) P(x)=-x^2+100x-600

Q: 2. The Minimum point of y= 320

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Q: A fertilizer producer finds that it can sell its product at a price of p = 2500 – 0.5x dollars per…

A: Given that p=2500-0.5x

Q: A firm can produce 400 units per week. If its total cost function is C = 300 + 1400x dollars and its…

A: Given Data: The number of units produced per week is: n=400 units per week The total cost function…

Q: A fertilizer producer finds that it can sell its product at a price of p = 2500-0.5x dollars per…

A: Given: p = 2500 - 0.5x  C(x) = 75000+625x+0.125x2

Q: A firm can produce 400 units per week. If its total cost function is C = 700 + 1500x dollars and its…

A: Given  total cost function is C = 700 + 1500x dollars total revenue function  R = 1900x − x2 dollars…

Q: A fertilizer producer finds that it can sell its product at a price of p = 1000 – 0.2x dollars per…

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Q: a certain company, the cost function for producing x items is C(x)=50x+200 and the revenue function…

A: Cost function for producing x items is given by- C(x)=50x+200 Revenue function for selling x items…

Q: Locate the critical points, and determine the maxima and minima. y = ( 2x - 1)2

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Q: Suppose that the price per unit in dollars of a cell phone production is modeled by p = $45 -…

A: Given:   The price per unit of phone is given as P=$45-0.0125x     where x is in thousands of phones…

Q: Assume the availability in the finishing and packaging departments are 350 and 125 hours…

A: Let x represents the regular model and y represents the catcher's model. From the given information,…

Q: A fertilizer producer finds that it can sell its product at a price of p = 1000 – 0.2x dollars per…

A: We need to find units to maximize profit.

Q: Suppose that the manufacturer of a gas clothes dryer has found that, when the unit price is p…

A: Given: -

Q: a company that makes and sells digital cameras has found that the total weekly cost of producing…

A: We have given the cost function C(x)=0.5x^2-210x+26250

Q: Find the relative maxima or minima. Maximum of ƒ(x, y) = x2 - 5y2, subject to x - y = 12

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Q: A fertilizer producer finds that it can sell its product at a price of p = 3500 – 0.7x dollars per…

A: Given that the selling price per unit is, p=3,500-0.7x dollars. The total production cost is,…

Q: Suppose that the selling price p of an item for the quantity x sold is given by the function 1 x +…

A: Given,  p=-16x+26

Q: A firm can produce 200 units per week. If its total cost function is C = 400 + 1300x dollars and its…

A: topic - application of derivatives

Q: The minimum value w=_____ occurs when y1=_____ y2=______, and  y3=________

A: To solve the Linear programming problem: minimize: w=14y1+56y2+64y3Such…

Q: Suppose that the manufacturer of a gas clothes dryer has found that, when the unit price is p…

A: Solution:GivenR(p)=-4p2+4000pTo find the unit price when revenue is maximum.Also find the Maximum…

Q: Locate the critical points and determine whether maxima, minima or inflection point. b. у %3D (x —…

A: Here we will find critical points and determine maxima, minima or inflection point,

Q: The cost of producing x boards is given by x2 + 20x + 500, while the price of each board if there is…

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Q: The manager of a 100 unit apartment complex is trying to decide what to charge for rent. Experience…

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Q: The marketing department at XYZ Company has found that, when Product ABC is sold at a price of p…

A: Given p = cost of the item x = sale of the item costing 'p'.  x = 280-14p R = Revenue, xp   To find:…

Q: A fertilizer producer finds that it can sell its product at a price of p = 1500 – 0.3x dollars per…

A: GIVEN: p = 1500 - 0.3x C(x) = 45000 + 375x + 0.075x2

Q: Given f (x)= (x^2) (e^-x) , find the relative maximum and relative minimum point respectively.

A: We will use derivative to find maximum and minimum value for the function. If f(x) is a function.…

Q: The demand function for a product is given by p+ 2x = 100 and the supply function is given by p- 3x…

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Q: At which value does the function g(x)= 3x^4 - 4x^3 attain a minimum and what is the minimum value?.

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Q: For the given cost function C(x) = 52900 + 400x + x2, First, find the average cost function. Use it…

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Q: Solve the problem graphically and calculate the Z value of at the optimal solution (

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Q: Find the value for x for which the graph of y =2e reaches its first minimum value.

A: To find the value for x for which the graph of below equation reaches its first minimum value.…

Q: A camera company estimates that the demand function for its new consumer digital camera is p=312e^…

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Q: A firm can produce 300 units per week. If its total cost function is C = 300 + 1200x dollars and its…

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Q: y = x* – 4x +1

A: