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If the hessian evaluated at a critical point is positive (negative) definite, then we can conclude that it's a local minimum (maximum) there. If the hessian is indefinite (both negative and positive eigenvalues), then it's a saddle point.
What happens if the Hessian is positive SEMI-definite?

gladilkamwy 2022-08-14

I am given a $3$ variable function:
$f (x, y, z) = \cos (x y) + \cos (y z) + \cos (z x)$
How to do the Hessian matrix for a $3$ variable function.

Trystan Castaneda 2022-08-13

For this linear system:
$\frac{d x}{d t} = x + y - 2$
$\frac{d y}{d t} = x - y - 4$
Classify $(1, 1)$

Carsen Patel 2022-08-12

Proposition: each saddle point is a isolated critical point.
Find a counterexample to disprove it

tuanazado 2022-08-09

Do have an example of a function $f : R^{2} \to R$ that has a critical point that is neither a saddle point nor a local (or global) extremum?

allucinemsj 2022-08-09

Find Critical point and nature
$4 x (x^{2} + y^{2} - 1) = 0$
$4 y (x^{2} + y^{2} + 1) = 0$

rivasguss9 2022-08-08

Why should the gradient be 0 at a critical point?

skilpadw3 2022-07-27

Determine the nature and stability of the critical point (0,0) for the following system:
$\frac{d x}{d t} = x + 2 y + 2 \sin y$
$\frac{d y}{d t} = - 3 y - x e^{x}$

Nathalie Fields 2022-07-24

Determine the nature and stability of thecritical point (0,0) for the following system:
$\frac{d x}{d t} = - \sin (x - y)$
$\frac{d y}{d t} = 1 - 5 y - e^{x}$

Haley Madden 2022-07-23

Determine the total number of critical points of the function $f (x) = (x + e^{x})^{k}$ , where $k > 0$ is an integer

Taniya Burns 2022-07-22

Suppose $f : R \to R$ has two continuous derivatives, has only one critical point $x_{0}$ , and that $f^{″} (x_{0}) < 0$ . Then $f$ achieves its global maximum at $x_{0}$ , that is $f (x) \leq f (x_{0})$ for all $x \in R$ .

makaunawal5 2022-07-21

In order to find the minimum/ maximum point of a function, I would take it's derivative and find critical points, points where the derivative is zero or undefined, and then put them on a number line and do strawberry field. If it's negative to the left of the point and positive to the right of the point, the point is a minimum, etc. When the denominator of your fraction is equal to zero, isn't that where you have a vertical tangent line and not a candidate for a min/max point?

Marisol Rivers 2022-07-20

Whats the difference between the critical point of a function and the turning point?Aren't they both just max/min points?

Aleah Booth 2022-07-15

Find the critical point of
$f (x) = \frac{5 x}{x - 3}$

Kenya Leonard 2022-07-14

What is the basic idea for finding critical point via Morse theory and critical groups?

A critical point is an important and special point on a graph of a function. It is associated with a local extremum (maximum or minimum) of the function, either in the x-direction or y-direction. It is also a point on the graph where the derivative of the function is either undefined or zero. In addition, the tangent line of the graph at the critical point is either parallel or perpendicular to the x-axis. Critical points are widely used in mathematics to solve a variety of problems such as solving equations, finding the area under a curve, and determining the global extrema of a function. Knowing the critical points of a function is essential for understanding the behavior of the function and its graph.

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