Calculator
Example Data Table
| Function | Interval | Main candidates | Expected result |
|---|---|---|---|
| x^3-3*x | [-2, 2] | x = -1, x = 1, endpoints | Maximum near x = -1, minimum near x = 1 |
| x^2-4*x+1 | [0, 5] | x = 2, endpoints | Minimum near x = 2, maximum at an endpoint |
| sin(x) | [0, 6.283185] | x near pi/2 and 3*pi/2 | Maximum near pi/2, minimum near 3*pi/2 |
Formula Used
For a closed interval [a, b], the calculator tests endpoint values f(a) and f(b). It also finds interior candidates where f'(x) = 0 or where the numerical derivative is close to zero.
The central difference estimate is f'(x) ≈ (f(x + h) - f(x - h)) / (2h). Each candidate value is compared. The largest value becomes the absolute maximum. The smallest value becomes the absolute minimum.
How To Use This Calculator
- Enter a function using x as the variable.
- Type the left and right interval endpoints.
- Set samples higher for fast changing functions.
- Choose decimal precision for the output table.
- Press the calculate button.
- Review the result above the form.
- Use the CSV or PDF button to save results.
Maxima Minima In An Interval Guide
What This Calculator Does
This calculator studies a function on a closed interval. It helps find the largest and smallest values that appear between two endpoints. The tool checks endpoints first. It then searches for critical points inside the interval. A critical point happens where the derivative is zero or nearly zero. Some functions also have sharp turns. For that reason, the calculator samples many points and tests nearby behavior.
Why Intervals Matter
A function may rise forever on an open line. A closed interval gives a clear start and end. That makes comparison possible. The extreme value theorem says a continuous function has an absolute maximum and minimum on a closed interval. This calculator follows that idea. It evaluates each valid candidate, then compares the results.
Helpful Inputs
Use x as the variable. Enter expressions like x^3-3*x, sin(x)+x/4, or exp(-x^2). Set the left and right endpoints. Choose more samples when the graph changes quickly. Use a smaller derivative step when you need more sensitive checks. A practical tolerance helps catch almost flat points without reporting too many false candidates.
Reading The Result
The result table lists each candidate point. It shows the x value, function value, source, and classification. Endpoint rows come from the interval limits. Critical rows come from derivative tests. The absolute maximum is the highest listed value. The absolute minimum is the lowest listed value. Local labels describe behavior near that point.
Good Practice
Start with a simple interval. Review the example table before using complex expressions. If a result looks unexpected, increase the sample count and compare again. Avoid undefined intervals, such as log values at negative inputs. Use parentheses carefully. The export buttons save the same result for records, homework checks, reports, or later review.
Limits And Checks
Numerical methods estimate derivative roots. They are very useful, but they are not symbolic proof. Smooth functions usually work well. Functions with jumps, holes, vertical asymptotes, or very narrow peaks need care. Always compare the table with your expected graph. For final school work, write the exact derivative when required. These checks make the answer easier to audit before sharing saved calculation files with others.
FAQs
What is a maximum in an interval?
It is the largest function value found between the interval endpoints, including the endpoints themselves.
What is a minimum in an interval?
It is the smallest function value found on the selected interval after comparing endpoints and critical points.
Does this calculator use exact symbolic derivatives?
No. It uses numerical derivative checks. This makes it flexible for many expressions, but final proofs may need symbolic work.
Which variable should I use?
Use x as the only variable. Other letters are treated as unknown names and may cause an error.
Are trigonometric values in degrees?
No. The calculator uses radians for sin, cos, tan, and inverse trigonometric functions.
Why should I increase samples?
Higher samples help find critical points in functions with sharp turns, waves, narrow peaks, or fast changes.
Can endpoints be maxima or minima?
Yes. Absolute extrema on closed intervals often occur at endpoints, so the calculator always checks them.
Why can undefined values affect results?
Discontinuities, division by zero, or invalid logarithms can hide candidates. Choose intervals where the function is defined.