Differentiation

Edexcel IAL WMA12/01 /P2/Oct 2019/Q1 (Applications of Differentiation)

A curve C has equation​​ y=2x2(x-5)

• Find, using, the x coordinates of the stationary Point of C.

(4)

• Hence find the values of​​ for which y is increasing.

(2)

SOLUTION

a-​​ 

(At stationary points, the gradient of the curve is equal to zero. Therefore, we will diffrentiate the equation of curve and equate the diffrentiated expression to zero. )

y=2x2x-5

y=2x3-10x2

dydx=6x2-20x

Now,​​ dydx=0,  

6x2-20x=0

Simplifying the equation

3x2-10x=0

x3x-10=0

x=0              3x-10=0

                        3x=10

                           x=103

b- ​​ Since, the question says ‘hence’, it means we have to use the answer we found in previous part to answer the ​​ question.​​ 

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

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

So we need to check that​​ f’’x​​ at the values of x= 0 and 10/3.​​ 

dydx=6x2-2x2

Finding second derivative

d2ydx2=12x-20

At​​ x=0

d2ydx2= -20<0 (maximum)

At​​ x=103

d2ydx2=12103-20=20>0 (minimum)

This can be seen in the sketch below as​​ 

Thus, values of x for which y is increasing are​​ 

x<0​​ and​​ x>103