Question
The graph below shows the derivative function 𝑓′ of a function f.
a. State the values of x for which f has stationary points and determine their nature.
b. Sketch a possible graph of f.
ABSOLUTE MAXIMA AND MINIMA (Ex 10F)
Question
From a square piece of metal of side length 2 m, four squares are removed and the metal is
then folded to give an open box with sides of height x metres.
a. Show that the volume of the box is given by: 𝑉(𝑥) = 4𝑥 3 − 8𝑥 2 + 4𝑥
b. Sketch the graph of V against x for a suitable domain.
c. If the height of the box must be less than 0.3 m, what will be the maximum volume of the
box?
MAXIMUM MINIMUM PROBLEMS (Ex 10G)
STEPS
1. You need to ensure that you understand the quantity that must be optimized (maximized or
minimized).
2. You need to express this quantity as a function f of a single variable (x). You must establish
the domain for this variable.
3. Differentiate the quantity to be maximized or minimized and let 𝑓 ′ (𝑥) = 0 for a stationary
point. Confirm that this value lies within the domain found in Step 2 above.
4. If the question asks you to “justify” that you have obtained a maximum/minimum, you need
to do a gradient chart to check the sign of the derivative either side of the stationary point.
5. Substitute the value of x obtained in Step 3 above back into the original function f to
establish the value of fmax
Question
A Queensland resort has a large swimming pool as illustrated with AB = 75 m and AD = 30m.
2
A boy can swim at 1 m/sec and run at 1 3 m/sec. He starts at A, swims to point P on DC, and
then runs from P to C. It takes him 2 seconds to pull himself out of the pool.
D
P
A
B
Let DP = x. Let T = total time taken for journey
3
a. i. Show that 𝑇 = √𝑥 2 + 900 + (75 − 𝑥) + 2
5
ii. What is the domain for the function T ?
b. Find
𝑑𝑇
𝑑𝑥
c. i. Find the value of x for which the time T will be a minimum. Justify
that you have found a minimum time.
ii. Find the minimum time.