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A critical point is a point on a graph at which the derivative is either equal to zero or does not exist.

If a critical point is equal to zero, it is called a stationary point (where the slope of the original graph is zero). If it does not exist, this can correspond to a discontinuity in the original graph or a vertical slope.

For functions of a single variable, critical points satisfy

$\dfrac{df}{dx} = 0.$

For functions of multiple variables, critical points satisfy

$\dfrac{\partial f(x_1,x_2,\ldots,x_n)}{\partial x_i} =0, \forall i \in \N \leq n.$

Properties

A critical point equal to zero may indicate the presence of an extreme value, if the second derivative of the function is non-zero. A positive second derivative indicates a local minima, and a negative second derivative indicates a local maxima. Note that some functions (e.g. $f(x,y)=x^3$ or $f(x,y)=y^2-x^2$) have critical points that aren't extreme values.

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