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Maxima and Minima

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Questions and Answers

What are the local or relative extrema of a function?

The maximum and minimum value of the function at a specific point

The sample maximum and minimum in statistics

The greatest and least elements in the set

The largest and smallest value taken by the function within a given range (correct)

What did Pierre de Fermat propose for finding the maxima and minima of functions?

An algorithm for calculating global extrema

A specific formula for determining local extrema

A general technique called adequality (correct)

A method for solving differential equations

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 $x \in X$ (correct)

$x_{0} \in X$ is a global maximum point if $f(x_{0}) = f(x)$ for all $x \in X$

$x_{0} \in X$ is a global maximum point if $f(x_{0}) < f(x)$ for all $x \in X$

$x_{0} \in X$ is a global maximum point if $f(x_{0}) > f(x)$ 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 Signup and view all the answers

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 Signup and view all the answers

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 $x$ Signup and view all the answers

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 $(a, b)$ Signup and view all the answers

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 $C = f_{yy}(a, b) < 0$ Signup and view all the answers

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 Signup and view all the answers

What are the points called at which both first-order partial derivatives of a function with respect to its variables are zero?

Critical points Signup and view all the answers