MATH1011 - Lecture 8

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MATH1011 - Lecture 7
Skeleton Slides

Maxima and Minima

• Review: Types of Stationary Points
• Apply your Year 11 differentiation techniques to find absolute and local extrema
  • e.g. 2nd derivative, critical points, etc.
• Remember for a function defined over an interval to check the value of $f(x)$ at both ends of the interval; they may be the absolute max/min

Maxima and Minima of Vector-Valued Functions

• A critical point of a vector-valued function occurs at any point where $\nabla f(\mathbf{c})=0$ or $\nabla f(\mathbf{c})$ does not exist
  • See: The Gradient

Saddle Point

• A critical point of $f$ is called a saddle point if $f$ has neither a local minimum nor a local maximum at the point

Boundary

• The boundary of a function are the ‘edges’ of a function

  • i.e. the boundary $\partial D$ of a subset $D$ is the set of all points such that every open disc centred at on the boundary contains points inside and outside the domain

    

  • The boundary is a subset of the domain

Boundary Cont.

• A subset $D$ is called closed if all its boundary points are in the subset, that is, if $\partial D\subseteq D$
• A subset $D$ is called open if none of its boundary points are in the subset, that is, if $\partial D\cap D=\varnothing$
• A set $D$ is called bounded if it is contained in a disc of finite radius

• Open and closed refer to $x$
• Bounded and unbounded refer to $y$

Extreme-Value Theorem

• For a function over $D$ that is a non-empty, closed and bounded subset and the function $f$ is a continuous function, there is an absolute maximum and minimum on $D$

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MATH1011 - Lecture 9
Annotated Slides