Example Data Table
| Function |
Interval |
Expected behavior |
Suggested samples |
| x^3 - 3*x |
-3 to 3 |
One local maximum and one local minimum |
500 |
| sin(x) |
0 to 6.28318 |
Wave peak and trough |
800 |
| -(x-2)^2 + 5 |
-2 to 6 |
Single maximum near x = 2 |
300 |
| sqrt(x) - x/4 |
0.1 to 16 |
Curved maximum inside interval |
700 |
Formula Used
The calculator evaluates y = f(xm)k + c. Here m is the input x multiplier,
k is the output y multiplier, and c is the output offset. It samples the
interval, finds likely turning points, and refines them with golden-section search.
A local maximum is found when nearby y values rise before the point and fall after it.
A local minimum is found when nearby y values fall before the point and rise after it.
The first derivative is estimated by a centered difference.
How to Use This Calculator
- Enter a valid expression using x as the variable.
- Set the start and end values for the graph interval.
- Choose sample count and decimal precision.
- Use conversion multipliers when the displayed units differ.
- Press Calculate to view extrema above the form.
- Use CSV or PDF download for saved reports.
Understanding Extrema on Graphs
A graph can rise, fall, pause, and turn. Maxima and minima describe those turning places.
They are useful in many conversion tasks. A converted curve may show the best output,
the lowest cost, or the safest range. This calculator samples a function across a chosen
interval. It then refines likely turning points with a numeric search. The result is a
practical estimate for real work.
Why Numeric Methods Help
Many formulas are hard to solve by hand. Some contain roots, powers, trigonometric parts,
or exponential growth. A symbolic method may be slow. A numeric method is flexible. It
evaluates the curve at many x values. It compares nearby y values. It then improves each
candidate point. This approach is useful when the expression comes from measured data or
a unit conversion model.
Reading the Results
The global maximum is the highest converted y value inside the interval. The global
minimum is the lowest converted y value inside the interval. Local maxima and minima are
turning points near their neighbors. Endpoints are also checked. They can be global winners
even when they are not turning points. The derivative estimate shows the slope near each
point. The second derivative estimate helps classify the curve shape.
Conversion Features
The input multiplier lets you convert the displayed x value before it enters the formula.
The output multiplier and offset convert the computed y value after evaluation. This is
helpful for unit changes, calibration, and scaled charts. Keep the output multiplier
positive when you want maximum and minimum labels to match the original curve direction.
Graphing Workflow
Start with a clean interval. Avoid huge ranges unless you need them. Increase samples for
curves with many waves. Use more precision for reports. Check the graph for breaks or
steep spikes. Export the table when you need spreadsheet review. Export the report when
you need a compact record. Always compare the result with domain knowledge. Numeric tools
estimate behavior, so sharp corners and undefined points need careful review. For best
accuracy, test smaller intervals around each candidate. If the same point appears again,
confidence improves. If results change a lot, increase samples and inspect the formula
syntax. Then save the final report securely.
FAQs
1. What functions can I enter?
You can enter expressions with x, numbers, powers, brackets, and common functions. Supported names include sin, cos, tan, sqrt, abs, ln, log, and exp.
2. Does this solve exact symbolic extrema?
No. It uses numeric sampling and refinement. This makes it flexible for many formulas, but results should be treated as estimates.
3. Why do endpoints appear in the result?
Endpoints can be the highest or lowest values on a closed interval. They may not be turning points, but they still matter for global extrema.
4. How can I improve accuracy?
Use more samples and a tighter interval around the point of interest. Smooth functions usually need fewer samples than fast oscillating functions.
5. What does the x multiplier do?
It converts displayed x before evaluation. For example, enter 1000 when displayed x is in kilometers and the formula expects meters.
6. What does the y multiplier do?
It scales the computed y value after evaluation. Use it to convert units or calibrate output values for reports and charts.
7. Why is my result missing?
The function may be undefined across much of the interval. Check divisions by zero, negative square roots, logarithms, and interval limits.
8. Can I download the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a compact report with key extrema and settings.