Extremum Command

This command differs among variants of English:

Extremum (US)
TurningPoint (UK + Aus)

Extremum( <Polynomial> ): Yields all local extrema of the polynomial function as points on the function graph.

Extremum(x³ + 3x² - 2x + 1) creates local extrema (0.29, 0.70) and (-2.29, 9.30) and shows them in the Graphics View.

Extremum( <Function>, <Start x-Value>, <End x-Value> ): Calculates (numerically) the extremum of the function in the open interval ( <Start x-Value>, <End x-Value> ).

Extremum((x⁴ - 3x³ - 4x² + 4) / 2, 0, 5) creates local extremum (2.93, -16.05) in the given interval and shows it in the Graphics View.

The function should be continuous in [ <Start x-Value>, <End x-Value> ], otherwise false extrema near discontinuity might be calculated.

CAS Syntax

Extremum( <Function> ): Will attempt to return all local extrema of the function (which should be continuous and differentiable)

Extremum(x³ + 3x² - 2x + 1) creates a list of the points and plots them $\left\{ \left(\frac{-\sqrt{15} - 3}{3}, \frac{10 \; \sqrt{15} + 45}{9} \right), \left(\frac{\sqrt{15} - 3}{3}, \frac{-10 \; \sqrt{15} + 45}{9} \right) \right\}$ .

Assume(0 < x < 20, Extremum(15/2 * sin( 2/15*pi * x) + 56/5)) yields the local turning points in the range given $\left\{ \left(\frac{15}{4}, \frac{187}{10} \right), \left(\frac{45}{4}, \frac{37}{10} \right), \left(\frac{75}{4}, \frac{187}{10} \right) \right\}$ .