10 Questions
What are the local or relative extrema of a function?
The largest and smallest value taken by the function within a given range
What did Pierre de Fermat propose for finding the maxima and minima of functions?
A general technique called adequality
How is a global maximum point defined for a real-valued function $f : X \to \mathbb{R}$?
$x_{0} \in X$ is a global maximum point if $f(x_{0}) \geq f(x)$ for all $x \in X$
In set theory, what are the maximum and minimum of a set defined as?
The greatest and least elements in the set
What does unbounded infinite sets, such as the set of real numbers, have in terms of minimum or maximum?
They have no minimum or maximum
What conditions need to be satisfied for a function $f(x, y)$ to have a relative maximum at $(a, b)$?
$f_x(a, b) = 0$ and $f_{xx}(a, b) > 0$, where $f_x$ and $f_{xx}$ are partial derivatives of $f(x, y)$ with respect to $x$
What is the nature of the extreme value at a point $(a, b)$ if both $A = f_{xx}(a, b)$ and $C = f_{yy}(a, b)$ are greater than zero?
The function has a minimum value at $(a, b)$
What are the conditions for a function to have neither a maximum nor a minimum value at a point $(a, b)$?
$A = f_{xx}(a, b) > 0$ and $C = f_{yy}(a, b) < 0$
What does it mean if the second-order partial derivative with respect to both variables is zero at a point $(a, b)$?
Further investigations are required to decide the nature of the extreme values
What are the points called at which both first-order partial derivatives of a function with respect to its variables are zero?
Critical points
Test your knowledge of finding the maximum and minimum values of functions, including local and global extrema, as well as the techniques proposed by Pierre de Fermat.
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