Critical Point Calculator
Enter Function
Use standard mathematical notation. Examples: x^2, sin(x), e^x, ln(x)
Function Analysis
Critical Points
No critical points found.
Visualization
Note on Limitations
This calculator works best with polynomial functions and simple trigonometric functions. For more complex functions, the critical points found may be incomplete. Always verify results for important calculations.
What are Critical Points?
Critical points of a function are points where the derivative equals zero or does not exist. These points are crucial in calculus as they help identify where a function might have local maxima, local minima, or saddle points.
How to Find Critical Points
To find critical points of a function f(x):
- Find the derivative f′(x) of the function
- Set the derivative equal to zero: f′(x) = 0
- Solve for x to find critical points
- Also check points where f′(x) does not exist
Types of Critical Points
Local Maximum
A point where the function value is greater than at nearby points. The second derivative is negative at these points.
Local Minimum
A point where the function value is less than at nearby points. The second derivative is positive at these points.
Saddle Point
A point that is neither a maximum nor a minimum. The second derivative equals zero and changes sign.
Applications of Critical Points
Critical points have numerous applications in various fields:
- Optimization problems in economics and engineering
- Finding maximum and minimum values in physics
- Analyzing stability in dynamical systems
- Determining points of inflection in curves