Maxima

And Minima Problems

Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state. (Plato)

The gradient of a curve at a point

𝑦𝑦 ′ > 0 leans right increasing ′′

Key concepts from curve sketching 𝑚𝑚

=

Gradient

𝑦𝑦 ′ > 0 leans right increasing

Concavity

The derivative of the related function evaluated at the point

𝑦𝑦 ′ < 0 leans left decreasing

𝑦𝑦 ′ < 0 leans left decreasing

𝑦𝑦 > 0 𝑦𝑦 ′′ > 0 𝑦𝑦 ′′ < 0 concave up concave up concave down Stationary point 𝒚𝒚′ = 𝟎𝟎 A point on the curve where the tangent is parallel to the 𝑥𝑥-axis ↔ the gradient is zero Nature of stationary points (Relative) (Relative) Maximum turning Minimum turning Horizontal point of inflexion point point 𝑦𝑦 < 0 concave down

′′

𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

𝑦𝑦 ′ = 0

𝑦𝑦 ′ = 0 𝑦𝑦 ′ = 0 𝑦𝑦 ′ = 0 What tells us the nature of the stationary points? These are the tests to justify your answers: 𝑦𝑦 ′′ < 0 𝑦𝑦 ′′ > 0 𝑦𝑦 ′′ = 0 𝑦𝑦 ′′ = 0 concavity changes about the point Before Before ′′ Ist derivative test 𝑦𝑦 < 0 𝑦𝑦 ′′ > 0 After After ′′ 𝑦𝑦 > 0 𝑦𝑦 ′′ < 0 Before 𝑦𝑦 ′ > 0 After 𝑦𝑦 ′ < 0

Before 𝑦𝑦 ′ < 0 After 𝑦𝑦 ′ > 0

Note: the gradient hasn’t changed Before 𝑦𝑦 ′ > 0 After 𝑦𝑦 ′ > 0

Prepared by Sue Millet for HSC Revision Day UOW

Before 𝑦𝑦 ′ < 0 After 𝑦𝑦 ′ < 0

Maxima and Minima Problems o Read the question, and annotate. (drawing any given diagrams as you read the question can help to improve understanding of the situation.) o Identify the variables (and any constants that are pretending to be variables.) o Focus on two variables –the quantity ( 𝑄𝑄 ) you want to maximise/minimise and one other variable ( 𝑥𝑥) -the question will help you to choose these variables. o Use the given information to reduce the variables -to eliminate a variable make it the subject of an equation then substitute for it. o Write an equation for the quantity you want to maximise/minimise in terms of your second variable ( 𝑄𝑄 = 𝑓𝑓(𝑥𝑥 )). The question will usually tell you this equation. o Simplify the equation, if this is possible and sensible. 𝑑𝑑𝑑𝑑 o Find the derivative ( 𝑑𝑑𝑑𝑑 ) 𝑑𝑑𝑑𝑑

o Find the stationary point value (𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑 = 0 𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥) o Test the stationary point value to establish the maximum/minimum using 

𝑑𝑑2 𝑄𝑄 𝑑𝑑𝑥𝑥 2

, if reasonable 𝑑𝑑𝑑𝑑

 sign of 𝑑𝑑𝑑𝑑 on each side of your solution, if the first derivative is too complex eg you already used the quotient rule o Read the question. o Answer the question. o Throughout the process use any given answers to work towards all or part of the solution. Curve Sketching Read the question, annotate and follow the directions in each part. These will generally require you to: 𝑑𝑑𝑑𝑑

o Find the derivative ( 𝑑𝑑𝑑𝑑 ) o Find the stationary points (𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑦𝑦′ = 0 𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥, 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒇𝒇𝒇𝒇𝒇𝒇𝒇𝒇 𝒚𝒚 𝑜𝑜𝑜𝑜 𝑖𝑖𝑖𝑖 𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑓𝑓(𝑥𝑥)) o Test the stationary points to establish their nature (maximum turning point/minimum turning point/ horizontal point of inflexion using  𝑦𝑦′′, if is reasonable  sign of 𝑦𝑦′ on each side of your solution, if the first derivative is too complex eg you already used the quotient rule o Find intercepts -definitely 𝑦𝑦 𝑤𝑤ℎ𝑒𝑒𝑒𝑒 𝑥𝑥 = 0, if examiners want 𝑥𝑥 𝑤𝑤ℎ𝑒𝑒𝑒𝑒 𝑦𝑦 = 0 they’ll usually ask. o Find points of inflexion 𝑦𝑦 ′′ = 0 if required.  You must test 𝑦𝑦 ′′ either side of the point to show concavity changes o Determine endpoints if given a domain. o Read the question. o Answer the question. o Remember the maximum/minimum value may be an endpoint value

Related skills o algebraic manipulation including confidence with expanding, factorising quadratics, indices, fractions, fractional indices,… o (Use of reference sheet for) differentiation, including application of the product, quotient and function of a function or chain rules

uv’+vu’

vu’-uv’ v2 derivative outer ×derivative inner

o these questions are often at the end of the exam targeting Band 6 but can also occur earlier and with good exam technique you should always be able to access some if not all of these marks

Have we learnt anything else about maximum and minimum values? The minimum value of a perfect square number/ expression is zero. Example: What is the range of 𝑦𝑦 = 𝑥𝑥 2 + 7 or 𝑦𝑦 = (𝑥𝑥 − 4)2 + 7?

Since the square term is always non-negative (≥0) 𝑦𝑦 ≥ 7 for both functions. In an exam I’d say

𝑥𝑥 2 ≥ 0 for all 𝑥𝑥 ∴ 𝑦𝑦 = 𝑥𝑥 2 + 7 ≥ 0 + 7 = 7 (𝑥𝑥 − 4)2 ≥ 0 for all 𝑥𝑥 ∴ 𝑦𝑦 = (𝑥𝑥 − 4)2 + 7 ≥ 0 + 7 = 7

This is called justifying your answer.

Have we learnt anything else about maximum and minimum values? The square root sign indicates only the positive square root Example: What is the range of 𝑦𝑦 = √𝑥𝑥 + 7

Since the square root term is non-negative (≥0) 𝑦𝑦 ≥ 7. In an exam I’d say √𝑥𝑥 ≥ 0 for all 𝑥𝑥 ∴ 𝑦𝑦 = √𝑥𝑥 + 7 ≥ 0 + 7 = 7 Another justify your answer.

So what about ? What is the range of 𝑦𝑦 = 7 − √𝑥𝑥

−√𝑥𝑥 ≤ 0 for all 𝑥𝑥 ∴ 𝑦𝑦 = 7 − √𝑥𝑥 ≤ 7 + 0 = 7

Have we learnt anything else about maximum and minimum values? The sine and cosine functions have maximum and minimum values −1 ≤ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≤ 1 and −1 ≤ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ≤ 1

Example: What is the maximum value of 4 − 5𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠? −1 ≤ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≤ 1 5 ≥ −5𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≥ −5 notice the sign reversal when you multiply by a negative. 9 ≥ 4 − 5𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≥ −1 −1 ≤ 4 − 5𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≤ 9 The maximum value is 9

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