**FINDING ZEROS AND CRITICAL POINTS**

Graphmatica can find approximate solutions (zeros) and extrema (critical points) for most Cartesian functions. (Since Newton's method of approximation is used to find these values, the function must be differentiable to find zeros and double differentiable to find critical points.) To calculate zeros and critical points, choose the **Find Critical Points **option from the **Calculus** menu. This will present you with a dialog box which allows you to do several things:

- Select an equation to use. The drop-down list allows you to select from all applicable equations on screen. Initially, the currently selected equation is chosen, if it is a Cartesian function. Otherwise the first applicable equation in the queue is chosen.
- Enter a guess to use as the seed for Newton's method. This may be anywhere within the domain of the function.
- Select whether you want to guess for zeros or extrema (maxima/minima).
- View the results. The listbox is divided up into three columns: type of point, solution, and value of the function at that point. The type of point is either
**Zero**,**Max**, or**Min**. The solution value is only printed for maxima or minima since for zeros it is (of course) always zero. If you have the**Point Tables**option on, the results will be logged to the Point Tables window as well.

When you initially select an equation, Graphmatica will automatically calculate and display all of the critical points **and** zeros it could find within the portion of the domain of the function which is currently on-screen. That is, values which lie outside the region currently on-screen will *not* be calculated. This is because the program makes guesses about potential critical points using information generated in the process of drawing a graph. To find more zeros or extrema outside the currently displayed region, either enter guesses manually or zoom out to collect guesses over a wider area.

**CAUTION**: Although Graphmatica will not allow you to use this feature when it can not find the first or second derivative of an equation, it cannot detect when one of these is not a smooth function. This may affect the accuracy of the results displayed. Consult a textbook for an explanation of why Newton's method is not accurate in this situation.

kSoft, Inc. ksoft@graphmatica.com Last updated: Sun 11 Jun 2017