Critical Values of Derivatives

critical_points(equation_type, coefficients, derivative_level, precision=4)

Calculates the critical points of a specific function at a certain derivative level

Parameters

• equation_type (str) – Name of the type of function for which critical points must be determined (e.g., ‘linear’, ‘quadratic’)

• coefficients (list of int or float) – Coefficients to use to generate the equation to investigate

• derivative_level (int) – Integer corresponding to which derivative to investigate for critical points (1 for the first derivative and 2 for the second derivative)

• precision (int, default=4) – Maximum number of digits that can appear after the decimal place of the results

Raises

• ValueError – First argument must be either ‘linear’, ‘quadratic’, ‘cubic’, ‘hyperbolic’, ‘exponential’, ‘logarithmic’, ‘logistic’, or ‘sinusoidal’

• TypeError – Second argument must be a 1-dimensional list containing elements that are integers or floats

• ValueError – Third argument must be one of the following integers: [1, 2]

• ValueError – Last argument must be a positive integer

Returns

points – Values of the x-coordinates at which the original function’s derivative either crosses the x-axis or does not exist; if the function is sinusoidal, then only five results within a two-period interval will be listed, but a general form will also be included; if the derivative has no critical points, then it will return a list of None

Return type

list of float or str

See also

• Roots for key functions: linear_roots(), quadratic_roots(), cubic_roots(), hyperbolic_roots(), exponential_roots(), logarithmic_roots(), logistic_roots(), sinusoidal_roots()

• Graphical analysis: sign_chart(), key_coordinates()

Notes

• Domain of a function: \(x_i = \{ x_1, x_2, \cdots, x_n \}\)

• Potential critical points of the derivative of the function: \(x_c = \{ c \mid c \in x_i, f'(c) = 0 \cup f'(c) = \varnothing \}\)

• Critical Points

Examples

Import critical_points function from regressions library

>>> from regressions.analyses.criticals import critical_points

Calulate the critical points of the second derivative of a cubic function with coefficients 2, 3, 5, and 7

>>> points_cubic = critical_points('cubic', [2, 3, 5, 7], 2)
>>> print(points_cubic)
[-0.5]

Calulate the critical points of the first derivative of a sinusoidal function with coefficients 2, 3, 5, and 7

>>> points_sinusoidal = critical_points('sinusoidal', [2, 3, 5, 7], 1)
>>> print(points_sinusoidal)
[5.5236, 6.5708, 7.618, 8.6652, 9.7124, '5.5236 + 1.0472k']