Inflection Point Calculator
- Free
- No sign-up
- Works on phones
- Shows the working
By Yasir Hafeez · Updated
This inflection point calculator finds where a function’s concavity changes. Enter \( f(x) \) and an interval, and it returns the second derivative, every inflection point as an \( (x, y) \) pair, and a table showing where the graph is concave up and where it is concave down.
It’s a quick way to check a curve-sketching problem or confirm the concavity intervals you found by hand. Free, no sign-up.
How to use the inflection point calculator
- Type your function. Use
^for powers,sqrt(x),sin(x),cos(x),ln(x),e^(x)andpi. Implicit multiplication like4x^3orx e^(-x)works. - Set the interval \( [a, b] \) to search. Every inflection point you care about must lie inside it.
- Press Analyze concavity. The output shows \( f''(x) \), the inflection points, and an interval table labeled “Concave up ∪” or “Concave down ∩”.
How it works
An inflection point is a point on the graph where \( f'' \) changes sign:
$$f''(x) > 0 \;\text{(concave up)} \quad\longleftrightarrow\quad f''(x) < 0 \;\text{(concave down)}$$
The calculator differentiates \( f \) twice symbolically, finds the zeros of \( f''(x) \) on \( [a, b] \), and keeps only those where \( f'' \) actually changes sign and \( f \) is defined. Those zeros split the interval into pieces; the sign of \( f'' \) at the midpoint of each piece decides its concavity.
Worked example
Find the inflection points and concavity of \( f(x) = x^4 - 4x^3 + 10 \) on \( [-2, 4] \).
Step 1. Differentiate twice:
$$f'(x) = 4x^3 - 12x^2, \qquad f''(x) = 12x^2 - 24x = 12x(x - 2)$$
Step 2. Solve \( f''(x) = 0 \): \( x = 0 \) or \( x = 2 \).
Step 3. Test the sign of \( f'' \) in each piece:
| Interval | Test point | \( f'' \) | Concavity |
|---|---|---|---|
| \( (-2, 0) \) | \( -1 \) | \( 36 \) | up |
| \( (0, 2) \) | \( 1 \) | \( -12 \) | down |
| \( (2, 4) \) | \( 3 \) | \( 36 \) | up |
Step 4. The sign changes at both zeros, so both are inflection points. With \( f(0) = 10 \) and \( f(2) = -6 \), the inflection points are \( (0, 10) \) and \( (2, -6) \).
Type the same function and interval into the calculator above to see this result. Notice that \( (0, 10) \) is also a critical point, since \( f'(0) = 0 \): the graph flattens out there without turning around.
Common mistakes
- Assuming \( f''(c) = 0 \) means inflection. For \( f(x) = x^4 \), \( f''(x) = 12x^2 \) is zero at 0 but positive on both sides, so there is no inflection point. The calculator only reports points where the sign really changes, so \( x^4 \) returns “No inflection points in this interval.”
- Forgetting where \( f \) is undefined. For \( f(x) = \frac1x \), \( f''(x) = \frac{2}{x^3} \) changes sign at 0, but 0 is not in the domain, so there is no inflection point, and the calculator lists none. To see the concavity on each side, run the two sides separately, for example \( [-2, 0] \) and \( [0, 2] \).
- Mixing up \( f' \) and \( f'' \). Critical points come from \( f' \); inflection points come from \( f'' \). To find where \( f' = 0 \), use the critical points method or the critical points calculator instead.
- Too small an interval. Only zeros of \( f'' \) inside \( [a, b] \) are searched. Widen the interval if you expect more points.
The hand method, with more examples, is in how to find inflection points.
Related guides
Further reading
- The Shape of a Graph, Part II (Paul’s Online Math Notes) — concavity, inflection points and the second derivative test.
- Higher Order Derivatives (Paul’s Online Math Notes) — practice finding the second derivative you need here.
FAQ
How do you find inflection points?
Compute \( f''(x) \) (the second derivative calculator can do this for you), solve \( f''(x) = 0 \) (and note where \( f'' \) is undefined), then check that \( f'' \) changes sign on either side of each candidate that is in the domain of \( f \).
What does concave up mean?
A graph is concave up where \( f''(x) > 0 \): it bends upward like a cup, and its slope is increasing. Concave down, where \( f''(x) < 0 \), is the opposite.
Can a function have no inflection points?
Yes. Parabolas, \( e^x \) and \( x^4 \) never change concavity, so they have none. The calculator then shows a single concavity for the whole interval.
Does the calculator give the y-coordinate too?
Yes. Each inflection point is listed as \( (x, f(x)) \), with exact x-values like \( \sqrt3 \) when they are recognized.