Critical Point Finder
Enter a polynomial function of x and find its critical points, classified as local minimums, local maximums, or neither.
How to use this calculator
Type a polynomial function of x. Use ^ for exponents, for example x^3 - 3x^2 + 2. Press Calculate.
This critical points calculator lists every critical point of your function. It labels each one as a local minimum, a local maximum, or neither. It also draws the graph, so you can see where the curve flattens out. Press Reset to start over.
What is a critical point?
A critical point is an x-value where the derivative of the function is zero or does not exist. A polynomial has a derivative everywhere, so its critical points are simply the places where f'(x) = 0.
At these points the slope of the curve is zero, so the graph is momentarily flat. Critical points are also called stationary points. Local peaks and valleys can only occur at critical points.
Three kinds of critical points
A local maximum is a peak. The function is higher there than at nearby points.
A local minimum is a valley. The function is lower there than at nearby points.
The third kind is neither. The curve flattens for a moment, then keeps going in the same direction. This is a stationary point of inflection. The point x = 0 on f(x) = x³ is the classic case.
How to find critical points step by step
2. Set f'(x) = 0 and solve for x. These x-values are the critical points.
3. Put each x-value into f(x) to get the y-coordinate of the point.
4. Classify each point with the second derivative test. If that test fails, use the first derivative test.
The second derivative test
Find the second derivative f''(x). Then evaluate it at each critical point.
| Value of f''(x) at the point | What it means |
|---|---|
| Positive | The curve bends upward. It is a local minimum. |
| Negative | The curve bends downward. It is a local maximum. |
| Zero | The test cannot decide. Use the first derivative test. |
The first derivative test
Check the sign of f'(x) just before and just after the critical point.
If f' goes from negative to positive, the point is a local minimum. If f' goes from positive to negative, it is a local maximum. If f' keeps the same sign on both sides, the point is neither.
This calculator applies both tests for you. It uses the second derivative test first. When f'' is zero, it checks the sign change of f' automatically.
Worked examples
f'(x) = 3x² − 6x = 3x(x − 2), so the critical points are x = 0 and x = 2.
f''(x) = 6x − 6. At x = 0, f'' = −6, which is negative, so (0, 2) is a local maximum.
At x = 2, f'' = 6, which is positive, so (2, −2) is a local minimum.
f'(x) = 4x³, so the only critical point is x = 0.
f''(x) = 12x², and f''(0) = 0. The second derivative test fails.
Now use the first derivative test. f' is negative for x < 0 and positive for x > 0. It changes from negative to positive, so (0, 0) is a local minimum.
f'(x) = 3x², so the only critical point is x = 0.
f''(0) = 0 again. But f' is positive on both sides of 0. There is no sign change, so (0, 0) is neither a maximum nor a minimum. It is a stationary point of inflection.
How to read the graph
The blue line is your function. Each critical point is a dot. Red dots are local maxima, green dots are local minima, and gray dots are points that are neither. The graph zooms to fit the critical points, so the shape around them is easy to see.
Common mistakes
Stopping at f'(x) = 0. Solving for x only gives candidates. You still need to classify each one.
Assuming f'' = 0 means neither. Compare x⁴ and x³ above. Both have f'' = 0 at x = 0, but one is a minimum and one is not.
Forgetting the y-coordinate. A critical point is a point on the graph. Put the x-value back into f(x).
Confusing local and absolute extrema. A local maximum is only the highest point nearby. It is not always the highest value of the whole function.
Frequently asked questions
What is the difference between a critical point and an extremum?
A critical point is any x-value where f'(x) = 0. An extremum is a critical point where the function really turns around. Every extremum is a critical point, but not every critical point is an extremum.
Why does the second derivative decide between a minimum and a maximum?
The second derivative tells you how the slope is changing. A positive value means the curve bends upward, which happens at a valley. A negative value means it bends downward, which happens at a peak.
How many critical points can a polynomial have?
A polynomial of degree n has at most n − 1 critical points. A cubic has at most two. A quartic has at most three. A polynomial can also have fewer, or none at all.
Can a polynomial have no critical points?
Yes. A linear function such as f(x) = 2x + 1 has a constant, nonzero derivative. It never flattens out, so it has no critical points.
What is a saddle point?
The term saddle point belongs to functions of two or more variables. For those functions, you find critical points by setting the partial derivatives equal to zero. In one variable, the closest idea is a stationary point of inflection, like x = 0 on x³. This tool works with one variable only.
Does this work for functions that are not polynomials?
No. Functions with trig, exponential or logarithmic terms need other differentiation rules, which this tool does not cover.