Critical Point

The critical points of a function are any points where its derivative is either zero, or is undefined. Points where the derivative is defined and is zero are known as stationary points.

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Definition §

Critical Point

Let $f$ be a function. Then a point $a$ is a critical point (of $f$) iff:

$$f^{\prime }(a)=0\text{ or undefined}$$

Where $f^{\prime }(a)$ is the derivative of $f$ at the point $a$. If $f^{\prime }(a)=0$, then $a$ is a stationary point. Else, it is a singular point.

Critical Point (Multivariate)

Let $f(x_1,\dots ,x_n)$ be a multivariate function. Then a point $P=(a_1,\dots ,a_n)$ is a critical point of $f$ iff:

$$\nabla f(a_1,\dots ,a_n)=\left(\frac{\partial f}{\partial x_1},\dots ,\frac{\partial f}{\partial x_n}\right){|}_{(a_1,\dots ,a_n)}=\vec{0}\text{ or undefined}$$

where $\nabla f(a_1,\dots ,a_n)$ is the gradient vector at point $P$. The same distinction between stationary points ($=\vec{0}$) and singular points (undefined gradient) apply.

Nature Of Critical Points §

Bivariate §

The nature of critical points for a bivariate function is given by the second partial derivative test.