• Stationary points occur when f′(x) = 0.
• Three types of stationary point exist and, by testing the sign of the gradient to the left and right of a stationary point, the nature (type) can be determined.
1. Local maximum turning points. ( f′(x) changes from + to − moving left to right.)
2. Local minimum turning points. ( f′(x) changes from − to + moving left to right.)
3. Points of inflection. ( f′(x) remains the same sign on both sides moving left to right.)
Solving maximum and minimum problems
• By solving the equation f′(x) = 0, and substituting the solutions into the original function, the maximum or minimum value of a quantity may be found. When the function is not provided it is necessary to formulate a rule in terms of one variable using the information given. Drawing a diagram to represent the situation is often useful.
• Always test to determine if a stationary point is a maximum or a minimum by checking the sign of the gradient to the left and right of the point.
• Check whether or not the local maximum or minimum is the absolute maximum or minimum. The absolute maximum or minimum may be the value of the function at one end of a specified interval.
Applications of antidifferentiation
• When the derivative of a function is known, antidifferentiation can provide the original function. Since the original function may have contained a ‘constant’, this must be allowed for, and can be found using the boundary conditions provided in the question.
summary
change in y change in x
---dy dx
---y
0 x
Local maximum
y
0 x
Local minimum
y
0 x
Point of inflection
f (x) y
0 a b x
Absolute maximum in the interval [a, b]
Local maximum
Multiple choice
1 The rate of change of f (x) = 2x3− 5x2+ 7 when x = 2 is:
2 If V = −3t2+ 7t + 50 then the average rate of change between t = 1 and t = 4 is:
3 If f (x) = 5 + 15x + 6x2− x3 then the gradient is zero when x equals:
4 The curve y = x2 − 10x + 21 has:
5 When x = −2, the graph of y = 2x2+ 3x − 5:
6 For a particular function g(x), g(1) = 0 and g′(x) < 0 if x ≥ 1. The graph which could represent g(x) is:
7 The maximum value of f (x) = −2x2+ 8x is:
8 The local minimum value of h(x) = x3+ 6x2− 28x − 3 occurs when x equals:
A −4 B 7 C −36 D 0 E 4
A −10 B −10 C −6 D −8 E 0
A 1 or −5 B 1 or 5 C −1 or 5 D −1 or −5 E 0 and −1
A a local maximum at (5, 0) B a local minimum at (5, −4) C a point of inflection at (5, 0) D a local maximum at (5, −4) E a point of inflection at (5, −4)
A is increasing B has a local maximum
C has a point of inflection D has a local minimum E is decreasing
A B C
D E
A 40 B 0 C 4 D −24 E 8
A 2 B –4 C 0 D −3 E 1
CHAPTER
review
9A
9A
23
---9B 9B
9B
9B
g(x) y
0 1 x
g(x) y
0 x 1
y
0 1 x
g(x)
g(x) y
0 1 x
g(x) y
0 x 1
9B
9B
13---9 The function g(x) = (x + 3) has:
10 A curve with a local maximum and a local minimum is:
11 The antiderivative of 12x + 3 is:
12 If the gradient of a curve is and its y-intercept is −3, then its rule is:
Short answer
1 If the position of a particle moving in a straight line is given by the rule x(t) = −2t2+ 8t + 3, where x is in centimetres and t is in seconds, find:
a the initial position of the particle
b the rate of change of displacement (that is, the velocity) at any time, t c the rate of change when t = 4
d when and where the velocity is zero
e whether the particle is moving to the left or to the right when t = 3 f the distance travelled in the first 3 seconds.
2 For the function f (x) = x3− 3x + 2:
a find the y-intercept b find the x-intercepts
c find the stationary points and state their type d sketch the graph of f (x).
3 If the volume of liquid in a vat, V litres, during a manufacturing process is given by V = 6t − t2, where t ∈ [0, 6], find:
a the rate of change 2 hours after the vat starts to fill up
b when the vat has a maximum volume.
4 If a piece of wire is 80 cm long:
a find the area of the largest rectangle that can be formed by the wire b determine whether a circle would give
a larger area.
5 Find the maximum possible volume of a fully enclosed cylindrical water tank given that the total internal surface area of the tank is 600 π square units.
A a local maximum when x = −3 B a point of inflection when x = −3 C a local minimum where x = −3 D a local minimum where x = 3 E a point of inflection where x = 3
A y = x3+ 2x2− 7x + 1 B y = x2− 3x + 1 C y = x3+ 7 D y = (x − 2)3 E y = x2+ 6x
A 6x2+ 3x B 24x2+ 3x + c C 24x2+ 3x
D 6x2+ 3x + c E 6x2
A y = x3+ x2− 10x − 3 B y = x3+ x2− 5x − 3 C y = x3+ 3x2− 10x − 3 D y = x3+ x2− 10x − 10 E y = x4− 10x2
9B
9C 9D
dy
9D
dx--- = (x–2) x 5( + )
1 3--- 3
2--- 1
3--- 3 2 ---1
3--- 3
2--- 1
4
---9A
9B
9C
9C
9C
6 The rate of increase of height, h metres, of an ascending helicopter at any time, t minutes after
it takes off is .
a Find an expression for the height at any time.
b Find the height 6 minutes after takeoff.
c Find the maximum height reached in the first 9 minutes.
7 A particle travels such that its velocity at any time, t, is given by v = 2t + 1.
a Given that velocity represents the rate of change of position, x, write down the relationship between v and x.
b If x = 3 when t = 2 write an expression for x in terms of t.
c Find the position of the particle when t = 10.
Analysis
1 A ball is thrown vertically up so that its height above the ground, h metres, at any time, t seconds, after leaving the
thrower’s hand is given by the function h(t) = t − t2+ 2.
a Find the height of the ball as it leaves the thrower’s hand.
b Find when and where the ball reaches its greatest height.
c Find when the ball returns to the same level that it left the thrower’s hand.
d If the ball isn’t hit, find when the ball hits the ground to the nearest
thousandth of a second.
e Hence state the domain and range of h(t).
f Sketch the graph of h versus t.
2 A piece of wire of length 100 cm is to be cut so that one piece is used to form a square, while the other is used to form a circle. If the edge length of the square is x cm, find, in terms of x, a the radius of the circle
b the area of the circle, and c the total area of the two shapes.
Show that, when x = 14, the total area is minimum.
h
9D
dh---dt = t2–14t+45
9D
9B
8 3--- 8
9
---9C
test test
CHAPTER
y
yourselfourself