Solving Cubic Polynomial Equation | Maxima And Minima | Graph

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A car dealership keeps track of how much it spends on advertising each month and of its monthly revenue. From this information, the list of advertising expenditures and probable associated revenues are shown in the table below.

a.    Find a cubic polynomial that models the data.

b.    Find the first derivative and explain what it means in terms of the data.

c.    Find the relative extremes and explain what they mean in terms of the problem.

d.    Find the second derivative and explain what it means in terms of the data.

e.    Find the point of diminishing returns and explain what it means in terms of the data.

1. a)The general cubic polynomial

y=ax³+bx²+cx+ d

There are four unknowns

Using (5,150) = 150= 125a +25 b +5c +d………………………………. (1)

Using (7,200) = 200 =343a +49b +7c +d………………………………... (2)

Using (9,250) =250=729a +81b+9c +d…………………………………… (3)

Using (11,325) = 325= 1331a +121b+11c+d…………………………... (4)

From equation 1 and 2 we get 218a +24b +2c =50……………… (5)

From equation 2 and 3 we get 386a +32b +2c =50……………… (6)

From equation 3 and 4 we get 602a +40b +2c=75………………….. (7)

On solving the above equation 5, 6, 7 we get;

a=0.52

b=-10.92

c=99.47

d=-139.06

Cubic polynomial equation y=0.52x³-10.92x² +99.47x -139.06

b)

y= 0.52x³-10.98x² +99.47x -139.06

Dy/dx = (0.52)3x²- 10.98(2x) +99.47 = 1.56x2 -21.96x+99.47

The first derivatives give the extrema of function and where the and where revenue is increasing or decreasing.

c)

To find maxima and minima dy/dx=0

1.5x² -21.96x+99.47 =0

X= (-b +-√b²- 4ac)/2a =1

X= 7.03 +3.771,   7.03 -3.771

Both of the equations are imaginary thus no maxima or minima is found.

d)

Second derivatives =d²y/dx²

d/dx (1.56x² -21.96x +99.47) =3.12x -21.96

If second derivative is positive at the point when first derivative is zero, it signifies minimum revenue.

If second derivative is negative at the point when first derivative is zero, it signifies maximum revenue.

e)

The point of diminishing according to the graph which was drawn;

Point of diminishing returns refers to part of graph just after maximum point.

f)

The Graph of function from part a.

Cubic equation is 436 - 100 x + 12.1x² - 0.347x³ 

Rosenbaum, R. A. Calculus: Basic Concepts and Applications. London: CUP Archive, 2015. print.

Spivak, Michael. Calculus. Texas: Cambridge University Press, 2014. print.