Critical Point
Calculus & Advanced MathA critical point is a location on a function where the derivative equals zero or is undefined, often corresponding to a peak, valley, or plateau.
Formula
f'(c) = 0 \text{ or } f'(c) \text{ undefined}
Definition
A critical point is a place on a curve where the slope is exactly zero (flat) or where the slope suddenly does not exist. These are the candidates for peaks and valleys.
Example
On a hill, the very top is flat for a moment before the slope drops again. That flat spot at the top is a critical point.
Key Insight
Finding critical points is the first step to locating the highest and lowest values of a function.
Definition
$c$ is a critical point of $f$ if $f'(c) = 0$ or $f'(c)$ does not exist. Every local extremum must occur at a critical point, but not every critical point is an extremum (some are inflection points with horizontal tangents).
Example
$f(x) = x^3$: $f'(x) = 3x^2 = 0$ at $x = 0$. But $x = 0$ is neither a max nor a min; it is an inflection point with a horizontal tangent.
Key Insight
Setting $f'(x) = 0$ finds critical points, but you must use the first or second derivative test to determine whether each one is a maximum, minimum, or neither.
Definition
In multivariable calculus, a critical point of $f: \mathbb{R}^n \to \mathbb{R}$ is where the gradient $\nabla f = 0$ or is undefined. Classification uses the Hessian matrix $H$: if $H$ is positive definite, the point is a local min; negative definite, a local max; indefinite, a saddle point.
Example
$f(x,y) = x^2 - y^2$: $\nabla f = (2x, -2y) = 0$ at the origin. The Hessian is $\begin{bmatrix} 2 & 0 \\ 0 & -2 \end{bmatrix}$, indefinite, confirming $(0,0)$ is a saddle point.
Key Insight
Critical points are the backbone of optimization in machine learning: gradient descent moves away from saddle points and toward local minima in a high-dimensional loss landscape.