Maximum and Minimum
The maximum and minimum of a function on a closed interval: the derivative, critical points and a graph.
f(x) =
x
from a =to b =
Examples: x^3 − 3x, sin x + cos x, x e^(−x), sqrt(4 − x^2), (x^2 + 4)/x. The interval ends are numbers: −1, 1/2, pi/2.
This calculator finds the maximum and minimum of a function on a closed interval step by step and draws the graph.
The method
- Find the derivative f′(x).
- Solve f′(x) = 0 and keep the roots inside the interval.
- Calculate the function at those points and at the ends of the interval.
- Choose the largest and smallest of these numbers.
Example. f(x) = x³ − 3x² + 2 on [−1, 3]. The derivative 3x² − 6x is zero at x = 0 and x = 2. The values are f(−1) = −2, f(0) = 2, f(2) = −2, f(3) = 2. The maximum is 2 and the minimum is −2.
For polynomials and fractions the points are exact. For other functions the equation is solved numerically, and approximate values are marked with ≈.
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