Differentiation

Edexcel IAL WMA12/01 /P2/June 2019/Q5 (Differentiation Applications)

A company makes a particular type of watch.​​ 

The annual profit made by the company from sales of these watches is modelled by the equation​​ 

P = 12x – x32 – 120 

where P is the annual profit measured in thousands of pounds and £x is the selling price of the watch.​​ 

According to this model,​​ 

• find, using calculus, the maximum possible annual profit.​​ 

(6)​​ 

• Justify, also using calculus, that the profit you have found is a maximum.​​ 

(2)

SOLUTION

a- ​​ To find the maximum possible annual profit, first, find the first derivative and then, equate the expression to zero. Second, using the value of x in the equation of P.​​ 

P=12x-x32-120

dpdx=12-32x12 

Now, ​​ local maximaum or stationary point,​​ dpdx=0 

dpdx=12-32x12

12-32 x12=0

24-3x12= 0

8-x12=0

x12=8

Squaring both sides to find x.​​ 

x=64

Now, substituting the value in equation of P.​​ 

P=12x-x32-120

P=1264-6432-120

P=136

Hence, the maximum profit is​​ £136.​​ 

b- Differentiating the first derivative of​​ y​​ to find double derivative and then putting the value of x as​​ 64​​ in the double derivated expression to find the nature of​​ y​​ at​​ x=64.

(If a function​​ f(x)​​ has a stationary point when​​ x=a, then if​​ f’’a<0, the point is a local maximum)

dpdx=12-32x12

Taking second derivative.

d2pdx2= -34 x12=-34x

when x=64 

d2pdx2= -3464

d2pdx2= -134 x 8 

d2pdx2= -332<0

Since​​ d2pdx2<0​​ when​​ x=64, this justifies the profit is maximum.