Extrema of Graph

extrema_points(equation_type, coefficients, precision=4)

Calculates the extrema of a specific function

Parameters

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

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

• 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 – Last argument must be a positive integer

Returns

• points[‘maxima’] (list of float or str) – Values of the x-coordinates at which the original function has a relative maximum; if the function is sinusoidal, then only two or three results within a two-period interval will be listed, but a general form will also be included; if the function has no maxima, then it will return a list of None

• points[‘minima’] (list of float or str) – Values of the x-coordinates at which the original function has a relative minimum; if the function is sinusoidal, then only two or three results within a two-period interval will be listed, but a general form will also be included; if the function has no minima, then it will return a list of None

See also

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

• Graphical analysis: critical_points(), sign_chart(), maxima_points(), minima_points(), key_coordinates()

Notes

• Critical points for the derivative of a function: \(c_i = \{ c_1, c_2, c_3, \cdots, c_{n-1}, c_n \}\)

• X-coordinates of the extrema of the function: \(x_{ext} = \{ x \mid x \in c_i, \left( f'(\frac{c_{j-1} + c_j}{2}) < 0 \cap f'(\frac{c_j + c_{j+1}}{2}) > 0 \right) \\ \cup \left( f'(\frac{c_{j-1} + c_j}{2}) > 0 \cap f'(\frac{c_j + c_{j+1}}{2}) < 0 \right) \}\)

• Minimum and Maximum Values

Examples

Import extrema_points function from regressions library

>>> from regressions.analyses.extrema import extrema_points

Calulate the extrema of a cubic function with coefficients 1, -15, 63, and -7

>>> points_cubic = extrema_points('cubic', [1, -15, 63, -7])
>>> print(points_cubic['maxima'])
[3.0]
>>> print(points_cubic['minima'])
[7.0]

Calulate the extrema of a sinusoidal function with coefficients 2, 3, 5, and 7

>>> points_sinusoidal = extrema_points('sinusoidal', [2, 3, 5, 7])
>>> print(points_sinusoidal['maxima'])
[5.5236, 7.618, 9.7124, '5.5236 + 2.0944k']
>>> print(points_sinusoidal['minima'])
[6.5708, 8.6652, '6.5708 + 2.0944k']