The Extreme Points Calculator helps locate possible local maxima and minima of a function. These points are important when studying where a graph changes from rising to falling or from falling to rising.
How Extreme Points Are Found
A common first step is to find critical points where fβ²(x)=0 or where fβ²(x) does not exist while the original function is defined. A critical point is a candidate for an extreme value, not proof that an extreme value occurs there.
Using the Calculator
Enter the function in the form expected by the tool and review the returned critical points or extrema. If a point is reported, check the function and derivative around that location to understand its behavior.
Example
Consider f(x)=xΒ²β4x+1. Its derivative is 2xβ4, which is zero at x=2. The second derivative is 2, so the graph is concave upward and the point at x=2 is a local minimum. The function value there is β3.
Local vs Global Extrema
A local maximum or minimum concerns nearby values. A global maximum or minimum is the greatest or smallest value over the full domain being considered. Endpoints and domain restrictions can matter when searching for global extrema on a closed interval.
Verify the Candidate
After locating a critical point, compare nearby function values or use an appropriate derivative test. This avoids treating every zero of fβ² as an automatic maximum or minimum.