WMA11 October 2021

10. WMA11/01 Edexcel​​ IAL P1 Oct 2021, Q10​​ (Integration & Differentiation: ​​ Second Derivative)

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

Given that​​ 

• fʹ(x) = ax – 12x13, where a is a constant​​ 

• fʹʹ(x) = 0 when x = 27​​ 

• the curve passes through the point (1, –8)​​ 

(a) find the value of a.​​ 

(3)

(b) Hence find f(x).​​ 

(4)

SOLUTION

a-​​ 

It is given fʹʹ(x) = 0 when x = 27, so find ing the second derivative of the function.​​ 

f'x=ax1-12x12

f''x=a-4x-23

0=a-4 27-23

a=42723

a=49

a=49

b-​​ 

To find the expression of f(x), integrate the f’(x) expression.​​ 

fx=∫f'x.dx

Where​​ f'x=49x-12x13

fx =∫49x -12x13dx 

fx=49x22-12x4343+c

fx=29x2-34 x 12x43+c

fx=29x2-9x43+C

Since the curve passes through the point (1, –8), it must satisfy the equation of the curve.​​ 

-8=29-9+c

1-29=c

c=79

So the equation of f(x) is

fx=29x2-9x43+79