Optimization Problems Calculator
Find maximum and minimum values of functions using calculus optimization techniques. Identify critical points and extrema with the first and second derivative tests.
Enter function using x. Examples: x^2, x^3-3*x^2+2, sin(x)*x
Theory & Formula
Optimization problems involve finding maximum or minimum values of functions. This is fundamental in calculus with applications in economics, engineering, physics, and more.
Kriitilised punktid
Critical points occur where f'(x) = 0 or f'(x) is undefined. These are candidates for local extrema.
\[f'(c) = 0 \text{ or } f'(c) \text{ does not exist}\]
Esimese tuletise test
- If f' changes from + to -, then f has a local maximum
- If f' changes from - to +, then f has a local minimum
Teise tuletise test
- \(f''(c) > 0\) ⇒ Local minimum (concave up)
- \(f''(c) < 0\) ⇒ Local maximum (concave down)
- \(f''(c) = 0\) ⇒ Test is inconclusive
Absolute Extrema
On a closed interval [a,b], absolute extrema occur either at critical points or at the endpoints.