Cambridge Additional Mathematics

GRAPHING

PACKAGE

y

y = f(x)

x

B

A,(-2 8)

C,(3 -11)

-4 5

O

378 Applications of differential calculus (Chapter 14)

Example 10 Self Tutor

Find and classify all stationary points of f(x)=2x^3 +3x^2 ¡ 12.

f(x)=2x^3 +3x^2 ¡ 12

) f^0 (x)=6x^2 +6x

=6x(x+1)

) f^0 (x)=0 when 6 x=0 or x+1=0

) x=0 or x=¡ 1

Also, f^00 (x)=12x+6

) f^00 (0) = 12(0) + 6 = 6 which is > 0

and f^00 (¡1) = 12(¡1) + 6 =¡ 6 which is < 0

So, we have a local minimum at x=0and a local maximum at x=¡ 1.

Now f(0) = 2(0)^3 + 3(0)^2 ¡12 =¡ 12

f(¡1) = 2(¡1)^3 +3(¡1)^2 ¡12 =¡ 11

) there is a local minimum at (0,¡12) and a local maximum at (¡ 1 ,¡11).

EXERCISE 14B

1 The tangents at points A, B, and C are horizontal.

a Classify points A, B, and C.

b Draw a sign diagram for:

i f(x) ii f^0 (x)

2 For each of the following functions, find and classify any stationary points. Sketch the function, showing

all important features.

a f(x)=x^2 ¡ 2 b f(x)=x^3 +1

c f(x)=x^3 ¡ 3 x+2 d f(x)=x^4 ¡ 2 x^2

e f(x)=x^3 ¡ 6 x^2 +12x¡ 7 f f(x)=

p

x+2

g f(x)=x¡

p

x h f(x)=x^4 ¡ 6 x^2 +8x¡ 3

i f(x)=1¡x

p

x j f(x)=x^4 ¡ 2 x^2 ¡ 8

3 At what value ofxdoes the quadratic function f(x)=ax^2 +bx+c, a 6 =0, have a stationary point?

Under what conditions is the stationary point a local maximum or a local minimum?

4 f(x)=2x^3 +ax^2 ¡ 24 x+1has a local maximum at x=¡ 4. Finda.

5 f(x)=x^3 +ax+b has a stationary point at (¡ 2 ,3).

a Find the values ofaandb.

b Find the position and nature of all stationary points.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_14\378CamAdd_14.cdr Monday, 7 April 2014 10:03:44 AM BRIAN