3.5.2

Maxima & Minima

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Turning Points of Graphs

The turning point of a graph is any coordinate on a graph where the gradient = 0. Another way to say this is that the derivative of the graph = 0 or that the gradient of the graph = 0.

Turning points

Turning points

  • Once you have found the derivative of a graph, f'(xx), you can find the xx coordinate of a turning point by making f'(xx) = 0.
  • When f'(xx) = 0, the rate of change of the function is equal to zero.
  • It is called a turning point because the graph often changes direction at f'(xx) = 0.
Example

Example

  • If f(xx) = xx2 − 2xx
    • f'(xx) = 2xx − 2
  • The turning point is when f'(xx) = 0
    • So, 0 = 2xx − 2
    • Solve for xx to find xx = 1
Example cont.

Example cont.

  • Find the yy-coordinate by substituting in the value of xx = 1 into f(xx).
    • f(1) = 12 − (2 × 1)
    • f(1) = −1
  • So the coordinate of the turning point of f(xx) = xx2 − 2xx is (1, −1).

Maxima & Minima

You can categorise turning points as a maximum or a minimum. Maxima and minima are the plurals of maximum and minimum.

Maxima & minima

Maxima & minima

  • A turning point is a local maximum if it is a high point when the graph is turning.
  • A turning point is a local minimum if it is a low point when the graph is turning.
The second derivative

The second derivative

  • You can find out if a turning point is a local maximum or local minimum using the second derivative.
  • The second derivative is found by differentiating the derivative.
    • We can write the second derivative of f(xx) is f''(xx)
    • We can write the second derivative of yy as d2ydx2\large\frac{d^2y}{dx^2} or yy''.
Local maximum or local minimum

Local maximum or local minimum

  • Once you have the xx value of the turning point, you can find out if it's a local maximum or a local minimum by substituting the xx value into the second derivative.
    • If the second derivative is less than 0, then it is a local maximum.
    • If the second derivative is greater than 0, then it is a local minimum.
Example

Example

  • f(xx) = xx3 + 6xx2 + 12 has a turning point at (0, 12). Is this a maximum or a minimum?
    • f'(xx) = 3xx2 + 12xx
    • And f''(xx) = 6xx + 12
    • Then substitute in the xx coordinate of the turning point into f''(xx).
    • f''(0) = (6 × 0) + 12 = 12
    • f''(xx) is positive at xx = 0 so the turning point at (0, 12) is a local minimum.
Jump to other topics
1

Numbers

1.1

Integers

1.1.1Integers1.1.2Ordering Numbers1.1.3The Four Rules of Operations1.1.4Hierarchy of Operations1.1.5Definitions & Terms1.1.6HCF & LCM1.1.7Standard Form1.1.8Standard Form 21.1.9Prime Numbers1.1.10End of Topic Test - Standard Form
1.2

Fractions

1.2.1Fractions1.2.2Using Fractions1.2.3Conversion1.2.4Reciprocals
1.3

Decimals

1.3.1Decimals1.3.2Decimals 21.3.3Exam-Style Questions - Recurring Decimals1.3.4Comparing & Converting Decimals1.3.5End of Topic Test - Fractions & Decimals
1.4

Powers & Roots

1.4.1Square Numbers1.4.2Index Notation1.4.3Exam-Style Questions - Index Notation1.4.4Advanced Indices1.4.5Surds1.4.6End of Topic Test - Powers & Roots
1.5

Set Language & Notation

1.5.1Sets
1.6

Percentages

1.6.1Percentages1.6.2Percentage Change1.6.3Percentages in Finance1.6.4Compound Interest
1.7

Ratio & Proportion

1.7.1Manipulating Ratios1.7.2Proportion
1.8

Degree of Accuracy

1.8.1Rounding & Significant Figures1.8.2Estimation1.8.3Upper & Lower Bounds1.8.4Limits of Accuracy1.8.5Limits of Accuracy 21.8.6End of Topic Test - Rounding & Estimation
2

Equations, Formulae & Identities

2.1

Algebraic Manipulation

2.1.1Algebra Basics2.1.2Power Rules & Collecting Terms2.1.3Expanding Brackets2.1.4Expanding Brackets 22.1.5Factorising Quadratics2.1.6Algebraic Fractions2.1.7Proofs2.1.8End of Topic Test - Algebra & Proofs2.1.9Exam-Style Questions - Algebraic Proof
2.2

Expressions & Formulae

2.2.1Rearranging Formulae2.2.2Simple Formulae2.2.3End of Topic Test - Algebra Manipulation & Formula
2.3

Linear Equations

2.3.1Solving Linear Equations2.3.2Simultaneous Linear Equations
2.4

Quadratic Equations

2.4.1Solving Quadratics2.4.2Solving Quadratics 22.4.3End of Topic Test - Solving Equations2.4.4Simulateous Quadratic Equations2.4.5Exam-Style Questions - Simultaneous Equations2.4.6Graphs of Simultaneous Equations2.4.7End of Topic Test - Simultaneous Equations
2.5

Proportion

2.5.1Graphs & Proportion
2.6

Inequalities

2.6.1Solving Inequalities2.6.2Graphs of Inequalities2.6.3Exam-Style Questions - Inequalities2.6.4End of Topic Test - Inequalities
3

Sequences, Functions & Graphs

3.1

Sequences

3.1.1Sequence Rules3.1.2Finding Terms3.1.3Exam-Style Questions - Sequences3.1.4End of Topic Test - Sequences
3.2

Functions

3.2.1Functions3.2.2Composite & Inverse3.2.3End of Topic Test - Functions
3.3

Graphs

3.3.1Plotting Graphs3.3.2Straight Line Graphs3.3.3Properties of Graphs3.3.4Quadratic Graphs3.3.5Practical Graphs3.3.6Graphs of Polynomials3.3.7End of Topic Test Graphs
3.4

Common Graphs

3.4.1Graphs of Common Functions3.4.2Transformations of Graphs3.4.3End of Topic Test - Transformations and Graph
3.5

Calculus

3.5.1Derivatives of Functions3.5.2Maxima & Minima
4

Geometry

4.1

Angles, Lines & Triangles

4.1.1Angles & Lines4.1.2Angle Rules4.1.3Angles in Triangles4.1.4Exam-Style Questions - Angles
4.2

Polygons

4.2.1Polygons4.2.2Quadrilaterals4.2.3Congruence4.2.4Scale Factor4.2.5Exam-Style Questions - Enlargements4.2.6End of Topic Test - Geometry
4.3

Symmetry

4.3.1Symmetry4.3.2Exam-Style Questions - Reflection
4.4

Measures

4.4.1Measurements4.4.2Time4.4.3Bearings4.4.4Speed
4.5

Circle Properties

4.5.1Properties of a Circle4.5.2Circle Theorems4.5.3Circle Theorems 24.5.4Circle Theorems 34.5.5Exam-Style Questions - Circle Theorems
4.6

Trigonometry & Pythagoras’ theorem

4.6.1Pythagoras' Theorem4.6.2Trigonometry4.6.3Sine & Cosine for Obtuse Angles4.6.4Triangle Formulae4.6.5Exam-Style Questions - Trigonometry & Area
4.7

Mensuration of 2D Shapes

4.7.1Standard Units4.7.2Perimeter of 2D Shapes4.7.3Area of 2D Shapes4.7.4Area & Circumference of Circle4.7.5End of Topic Test - Trigonometry & Shapes
4.8

3D Shapes & Volume

4.8.13D Shapes4.8.2Surface Area4.8.3Volume4.8.4Exam-Style Questions - Volume
5

Vectors & Transformation Geometry

5.1

Vectors

5.1.1Vectors5.1.2Vectors 25.1.3Exam-Style Questions - Vectors
5.2

Transformation Geometry

5.2.1Transformations5.2.2Rotations5.2.3Translations & Reflections5.2.4Enlargements5.2.5Effects of Transformations
6

Statistics & Probability

6.1

Statistical Measures

6.1.1Basic Statistics6.1.2Frequency Table
6.2

Graphical Representation of Data

6.2.1Pictograms, Bar Charts & Pie Charts6.2.2Histograms6.2.3Cumulative Frequency6.2.4Cumulative Frequency 26.2.5Exam-Style Questions - Cumulative Frequency6.2.6End of Topic - Datasets
6.3

Probability

6.3.1Probability6.3.2Theoretical Probability6.3.3Experimental Probability6.3.4Venn Diagrams6.3.5Sample Space Diagrams6.3.6Tree Diagrams6.3.7Independent Events6.3.8Conditional Probability6.3.9End of Topic Test - Probability6.3.10Exam-Style Questions - Probability

Practice questions on Maxima & Minima

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  1. 1
    How do you find the second derivative of a function?Multiple choice
  2. 2
    What will the value of the second derivative be at a turning point if the turning point is a local maximum?Multiple choice
  3. 3
    What will the value of the second derivative be at a turning point if the turning point is a local minimum?Multiple choice
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Derivatives of Functions

Angles & Lines