Critical Point Calculator – Find Maxima, Minima, Inflection Points
Find critical points, maxima, minima using derivatives
Table of Contents
How to Use
- Enter your function using polynomial notation (e.g., x^3 - 3*x^2 + 2*x)
- Use * for multiplication and ^ for exponents
- Click calculate to find all critical points
- Review the type of each point (minimum, maximum, or inflection)
What are Critical Points?
Critical points of a function are points where the derivative is zero or undefined. These points are important because they often correspond to local maxima, local minima, or inflection points.
To find critical points, we solve f'(x) = 0 for x. Then we use the second derivative test to determine the nature of each critical point.
Second Derivative Test
The second derivative test helps classify critical points:
- If f''(x) > 0 at a critical point, the point is a local minimum
- If f''(x) < 0 at a critical point, the point is a local maximum
- If f''(x) = 0, the test is inconclusive (may be an inflection point)
- Use the first derivative test as an alternative when the second derivative test fails
Steps to Find Critical Points
- Calculate the first derivative f'(x)
- Solve f'(x) = 0 to find candidate points
- Calculate the second derivative f''(x)
- Evaluate f''(x) at each critical point
- Classify each point based on the sign of f''(x)
- Calculate the y-coordinate by evaluating f(x) at each critical x
Applications
- Optimization problems: finding maximum profit or minimum cost
- Physics: analyzing motion and finding extremes of potential energy
- Economics: determining optimal production levels
- Engineering: optimizing design parameters
- Data analysis: finding peaks and troughs in data
Frequently Asked Questions
- What's the difference between local and global extrema?
- Local extrema are the highest or lowest points in a neighborhood around them. Global extrema are the absolute highest or lowest points over the entire domain. A local maximum might not be the global maximum.
- Can a function have no critical points?
- Yes. Linear functions (like f(x) = 2x + 3) have constant derivatives and no points where the derivative equals zero. Monotonically increasing or decreasing functions may have no critical points.
- What if the second derivative is zero?
- When f''(x) = 0, the second derivative test is inconclusive. You should use the first derivative test instead, checking how f'(x) changes sign around the critical point.