Calculator Inputs
Example Data Table
| Function | Lower Bound | Upper Bound | Expected Minimum x | Expected Minimum Value |
|---|---|---|---|---|
| x^2 - 6*x + 10 | -10 | 10 | 3 | 1 |
| (x-5)^2 + 4 | 0 | 12 | 5 | 4 |
| x^2 + 2*x + 8 | -8 | 4 | -1 | 7 |
Formula Used
This calculator uses the golden section search method. It works on a chosen interval. The method compares two inner points and removes the side that cannot contain the smaller value.
The golden ratio constant is: r = (√5 - 1) / 2
Inner points are calculated as: c = b - r(b - a) and d = a + r(b - a)
If f(c) < f(d), the new interval becomes [a, d]. Otherwise, the new interval becomes [c, b].
How to Use This Calculator
- Enter a function using x as the variable.
- Use operators such as +, -, *, /, and ^.
- Enter the lower and upper search bounds.
- Set a small tolerance for more accurate results.
- Choose the maximum number of iterations.
- Press the submit button to calculate the minimum.
- Review the result shown above the form.
- Download the result as CSV or PDF when needed.
About Function Minimization
Purpose
Function minimization helps locate the smallest output of a function within a selected interval. It is useful in algebra, calculus, engineering, economics, design, and data analysis. Many problems ask for the lowest cost, shortest distance, least error, or best setting. This calculator gives a practical way to estimate that point without solving derivatives by hand.
Search Method
The calculator uses golden section search. This method is designed for single variable functions. It works best when the function has one clear minimum inside the interval. The method does not require a derivative. It only needs function values. That makes it helpful for functions that are difficult to differentiate.
Input Control
The lower and upper bounds define the search range. A wider range may need more iterations. A narrow range can be faster. Tolerance controls when the search stops. Smaller tolerance gives a tighter answer, but it may require more steps. Maximum iterations protect the calculator from endless processing.
Result Meaning
The result includes the estimated x value and the related function value. The x value is the point where the function is lowest inside the chosen interval. The function value is the minimum output found by the search. The final bounds show the narrowed interval around the answer.
Accuracy Notes
Good bounds are important. If the true minimum is outside the interval, the answer will not represent the full function. If the function has many dips, the method may find a local minimum. For difficult functions, test several intervals. Compare results before making decisions.
Useful Expressions
You can enter expressions like x^2, sin(x), cos(x), sqrt(x), exp(x), log(x), abs(x), and pow(x,2). Use multiplication signs clearly. Write 2*x instead of 2x. Avoid unsupported symbols. Keep the expression simple and valid for every x value inside the selected interval.
FAQs
What does this calculator minimize?
It minimizes a single variable function over the interval you enter. It estimates the x value where the function gives its smallest output.
Does it use derivatives?
No. It uses golden section search. The method compares function values and narrows the interval step by step.
What type of functions work best?
It works best for continuous functions with one clear minimum inside the selected interval. Smooth algebraic and trigonometric expressions often work well.
Can it find global minimums?
Only inside the entered interval. If the function has several local minimums, test multiple intervals and compare the resulting function values.
Why do I need lower and upper bounds?
The method searches only between those bounds. Good bounds help the calculator locate the intended minimum faster and more reliably.
What does tolerance mean?
Tolerance is the stopping limit for interval width. A smaller tolerance usually gives a more precise result but may require more iterations.
Can I use functions like sin and sqrt?
Yes. Supported functions include sin, cos, tan, sqrt, log, exp, abs, pow, min, and max. Use x as the variable.
Why might I get an error?
An error may appear if the expression has unsupported symbols, invalid syntax, or produces undefined values within the selected interval.