A Company’s profit from a certain product is represented by p(x)= -5x²+1 , 125x-5000, where x is the price of the product . Compare the growth in profits from x= 120 to 140 and from x= 140 to 160 , What do you notice ?

The correct answer is: p(x) = -5x2 + 1 is a quadratic function and p(x) = 125x - 5,000 is a linear function.

Step-by-step solution:-
A). p(x) = -5x² + 1

Put x = 120- p(x) = -5(120)² + 1 = -71,999
Put x = 130- p(x) = -5(130)² + 1 = -84,499
Put x = 140- p(x) = -5(140)² + 1 = -97,999
Put x = 150- p(x) = -5(150)² + 1 = -1,12,499
Put x = 160- p(x) = -5(160)² + 1 = -1,27,999

∴ we notice that the growth (negative growth) in profits-

from x = 120 to 140 = -97,000 - (-71,999) = -97,999 + 71,999 = -26,000
from x = 140 to 160 = -1,27,999 - (-97,999) = -1,27,999 + 97,999 = -30,000

Hence, when price of the given product increases from 120 to 140, the profits decrease by 26,000 whereas when the price increasesfrom 140 to 160, the profits decrease by 30,000.
First difference-
                                                                    d₁ = p(130) - p(120) = -84,499 - (-71,999) = -12,500
                                                                    d₂ = p(140) - p(130) = -97,999 - (-84,499) = -13,500
                                                                   d₃ = p(150) - p(140) = -1,12,499 - (-97,999) = -14,500
                                                                 d₄ = p(160) - p(150) = -1,27,999 - (-1,12,499) = -15,500
Second difference-
                                                                                d₂ - d₁ = -13,500 - (-12,500) = -1,000
                                                                                d₃ - d₂ = -14,500 - (-13,500) = -1,000
                                                                                d₄ - d₃ = -15,500 - (-14,500) = -1,000
Since, the second difference for the given function is constant i.e. -1,000, the given function is a quadratic function.
For every increase in the price, the profits will fall increasingly.
B). p(x) = 125x - 5,000
                                                                         Put x = 120- p(x) = 125(120) - 5,000 = 10,000
                                                                         Put x = 130- p(x) = 125(130) - 5,000 = 11,250
                                                                         Put x = 140- p(x) = 125(140) - 5,000 = 12,500
                                                                         Put x = 150- p(x) = 125(150) - 5,000 = 13,750
                                                                         Put x = 160- p(x) = 125(160) - 5,000 = 15,000
∴ we notice that the growth in profits-
                                                                        from x = 120 to 140 = 12,500 - 10,000 = 2,500
                                                                        from x = 140 to 160 = 15,000 - 12,500 = 2,500
Hence, when price of the given product increases from 120 to 140, the profits increase by 2,500 and when the price increases from 140 to 160, the profits increase by 2,500.
First difference-
                                                                       d₁ = p(130) - p(120) = 11,250 - 10,000 = 1,250
                                                                       d₂ = p(140) - p(130) = 12,500 - 11,250 = 1,250
                                                                       d₃ = p(150) - p(140) = 13,750 - 12,500 = 1,250
                                                                       d₄ = p(160) - p(150) = 15,000 - 13,750 = 1,250
Since, the first difference for the given function is constant i.e. 1,250, the given function is a Linear function.
For every increase in the price, the profits will increase at a steady (constant) rate.
Final Answer:-
∴ p(x) = -5x₂ + 1 is a quadratic function and p(x) = 125x - 5,000 is a linear function.