WMA11 June 2022

• WMA11/01 Edexcel​​ IAL P1 June 2022, Q7​​ (Differentiation​​ and​​ Integration)

The curve C has equation y = f(x), x >​​ 0​​ 

Given that​​ 

• f'x=2x +Ax2+ 3,​​ where A is a constant​​ 

• f′′(x) = 0 when x = 4​​ 

• find the value of A.​​ 

(4)

Given also that​​ 

• f(x) =​​ 8 3, when x = 12​​ 

• find f(x), giving each term in simplest form.​​ 

(5)

SOLUTION​​ 

a- Finding the second derivative to equate the expression to zero to find the value of A.​​ 

f'x=2x-12 +Ax-2+3

f''x= -x-32-2Ax-3 

f''x= -1x3-2Ax3

As given in the question,​​ f''x=0​​ at​​ x=4.

f''4=0

-143-2A43=0

-18-2A64=0

-18=2A64

A=-18 x 32

A= -4

b-​​ Since integration is anti-derative, so to find​​ f(x)​​ lets integrate​​ f’(x)​​ expression.​​ ​​ 

f'x=2x -4x2+3

f'x=2x-12-4x-2+3

Since​​ fx=∫f'xdx, therefore integrating now.

fx= ∫2x-12-4x-2+3dx

fx=2x12 12-4x-1-1+3x+c 

fx=4x12+4x-1+3x+c

fx=4x+4x+3x+c

As given in the question, f(x) =​​ 8 3, when x = 12.

83=412+412+312+c

83=443+13+36+c

83=83+3613+c

C=-3613  

C=-1083

Thus, on putting the value of C in f(x) function back.​​ 

fx=  4x+4x+3x-1083