Critical Points of a Polynomial Calculator
The critical points of x³ − 3x + 2 are −1 and 1, where its derivative 3x² − 3 equals zero. Enter the coefficients, highest power first, and get the derivative and a sorted list of every real critical point. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Coefficients
- 1, 0, -3, 2
- Polynomial degree
- 3
- Derivative coefficients
- 3, 0, -3
- Critical points
- -1, 1
- Quantity
- 2
The critical points of a polynomial calculator differentiates a polynomial and finds every real x value where the derivative equals zero. Enter the coefficients from the highest power down to the constant term, keeping a zero wherever a power is missing. The result includes the derivative coefficients, the polynomial degree, and a sorted list of critical points. It is designed for checking calculus work, locating possible turning points, and producing structured results for automated analysis without relying on symbolic notation.
Enter the polynomial as an ordered coefficient list
Write the coefficients in descending power order, beginning with the coefficient of the highest power and ending with the constant. For example, x cubed minus 3x plus 2 is entered as [1, 0, -3, 2]. The zero is essential because it holds the place of the missing squared term.
The first meaningful coefficient must be nonzero, although harmless leading zeros are removed automatically. At least two values are needed because a constant polynomial does not have a finite list of isolated critical points: its derivative is zero everywhere. The calculator accepts up to twelve coefficients, so the original polynomial can have degree as high as eleven.
Each coefficient must be a finite number with an absolute value no greater than one trillion. Integers and decimals are both supported, as are negative coefficients and explicit zeros for any missing powers.
Understand how derivative roots become critical points
For a polynomial f, an interior critical point occurs wherever f'(x) is zero or undefined. Polynomial derivatives are defined for every real x, so this calculator only needs to solve f'(x)=0. It first applies the power rule term by term and returns the resulting coefficient list.
It then locates all real zeros of that derivative. The numeric solver recursively uses roots of successive derivatives to divide the real line into intervals where the polynomial is monotonic. A linear polynomial has a nonzero constant derivative and therefore returns an empty list.
Multiple critical points can appear when the derivative has several distinct real roots.
Use the results correctly and recognize the limits
A critical point is a candidate for a local maximum, local minimum, or stationary inflection point, but solving f'(x)=0 alone does not classify it. To determine its behavior, inspect the sign of the derivative on either side, apply an appropriate derivative test, or evaluate higher derivatives. The calculator deliberately reports only the x values requested, together with derivative coefficients that make the calculation auditable.
Numeric root finding also means that irrational answers are returned as decimal approximations rather than radicals or symbolic expressions. Within the declared coefficient and degree limits, the procedure is deterministic: it uses no network, random source, saved state, or current time, so identical input produces identical output. This makes it useful for homework checks and teaching examples, but also for data pipelines that need machine-readable candidate turning points.
What you can do with it
Check a calculus exercise
Compare your differentiated polynomial and critical x values with a deterministic structured result.
Locate candidate turning points
Find the real derivative roots that can be tested as local maxima, minima, or stationary inflection points.
FAQ
How should I order the coefficients?
List them from the highest power to the constant term, using zero for every missing intermediate power.
Does the calculator return complex critical points?
No. Critical points for a real-variable polynomial are reported as real x values only.
Why does a linear polynomial return an empty list?
Its derivative is a nonzero constant, so there is no x value where that derivative equals zero.
Does every returned point represent a maximum or minimum?
No. A derivative root can also be a stationary inflection point. Use a sign test or higher derivatives to classify it.
Does it give the y-values of the critical points?
No, it returns the x-coordinates only, sorted, plus the derivative used to find them. To get the height of each point, evaluate the original polynomial at those x-values; for x³ − 3x + 2 that means plugging in −1 and 1.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/calculus/critical-points-polynomial \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"coefficients":[1,0,-3,2]}'const res = await fetch("https://api.kit.forhosting.com/calculus/critical-points-polynomial", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"coefficients": [
1,
0,
-3,
2
]
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/calculus/critical-points-polynomial",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"coefficients": [
1,
0,
-3,
2
]
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/calculus/critical-points-polynomial", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"coefficients":[1,0,-3,2]}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"coefficients":[1,0,-3,2]}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/calculus/critical-points-polynomial", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"coefficients": [
1,
0,
-3,
2
]
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "calculus.critical_points_polynomial",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Enter the polynomial as an ordered coefficient list
This representation avoids ambiguity about variables, exponent symbols, multiplication signs, and spacing, making the same input dependable in both the browser and an API workflow.
Understand how derivative roots become critical points
A standard root bound supplies finite outer endpoints, sign changes are refined by bisection, and derivative-root checks preserve repeated roots that merely touch the axis without crossing it. The returned critical_points array contains x coordinates only; it does not include the corresponding y values.
Use the results correctly and recognize the limits
It does not solve complex roots, impose a closed interval, add interval endpoints, evaluate f(x) at each point, or draw a graph. Those are separate steps with different input requirements. API requests cost $0.002, while the browser execution can provide an immediate interactive check.
Integration use cases
Automate polynomial analysis
Send coefficient arrays through the API and reuse sorted critical points in a larger mathematical workflow.
Technical questions
What does an API request cost?
Each API request costs $0.002. Direct browser execution is available for interactive calculations.