Understanding Local Minima
A local minimum is a low point within a nearby region. The function may rise on both sides of that point. It does not need to be the lowest value across the whole interval. It only needs to be lower than close neighboring values.
Why Local Minima Matter
Local minima appear in design, finance, science, and optimization tasks. They can show the least cost near a decision. They can mark the most stable state of a system. They can also reveal where a curve stops falling and begins rising again.
How the Calculator Searches
This tool samples the selected interval first. It checks each valid point against its close neighbors. When a possible low point is found, the calculator refines it. Golden section search is then used inside a small bracket. This improves the x value without needing a symbolic derivative.
Derivative Checks
The calculator estimates the first derivative near each candidate. A value near zero suggests a stationary point. It also estimates the second derivative. A positive second derivative supports a local minimum. A value close to zero may mean a flat bottom or a higher-order turning point.
Choosing Better Inputs
Use a wide interval when you do not know where the low point sits. Use more scan points for curves with waves, sharp turns, or repeated valleys. Use a smaller tolerance when you need more decimal accuracy. Use a reasonable derivative step, because very tiny steps can magnify rounding error.
Reading the Output
The x column gives the estimated location. The f(x) column gives the function value at that point. The derivative columns help judge confidence. The classification explains the numerical evidence. Always review the interval and function domain before accepting a result.
Practical Notes
Numerical methods are estimates, not formal proofs. Discontinuous functions may give misleading candidates. Functions with undefined sections can be skipped during scanning. For important engineering, financial, or research work, compare the result with graphing and analytic methods.