Concavity Calculator
Evaluates the second derivative of a cubic f(x) = ax³ + bx² + cx at a test point to determine concavity. Use it to classify intervals as concave up or concave down and locate inflection points.
About this calculator
For a cubic polynomial f(x) = Ax³ + Bx² + Cx + D, the second derivative is f″(x) = 6A·x + 2B. This calculator evaluates f″ at a chosen test point x₀ using the formula: f″(x₀) = 6A·x₀ + 2B. If f″(x₀) > 0, the curve is concave up (bowl-shaped) at that point; if f″(x₀) < 0, it is concave down (hill-shaped). The inflection point, where concavity changes, occurs where f″(x) = 0, giving x_inflection = −B / (3A). At an inflection point the curve transitions from one concavity to the other, which is important in optimization and curve sketching. Note that coeffC (coefficient of x) affects f′ but not f″, so it does not appear in the concavity formula.
How to use
Let f(x) = 2x³ − 3x² + x. Enter Coefficient of x³ = 2, Coefficient of x² = −3, Coefficient of x = 1, Test Point = 1. Compute: f″(1) = 6·2·1 + 2·(−3) = 12 − 6 = 6. Since 6 > 0, the curve is concave up at x = 1. To find the inflection point: x = −(−3) / (3·2) = 3/6 = 0.5. So concavity switches at x = 0.5, and the curve is concave up for x > 0.5.
Frequently asked questions
How do you determine concavity using the second derivative test?
Compute the second derivative f″(x) of your function and evaluate it at a point x₀ inside the interval you want to test. If f″(x₀) > 0, the function is concave up on that interval; if f″(x₀) < 0, it is concave down. A result of exactly zero suggests a possible inflection point but requires further analysis — the concavity might not actually change. This method is most straightforward for polynomial functions where differentiation is mechanical.
What is an inflection point and how is it different from a local extremum?
An inflection point is where the concavity of a function changes from up to down or vice versa; it is identified by setting f″(x) = 0 and confirming a sign change. A local extremum (maximum or minimum) is where f′(x) = 0 and f″(x) ≠ 0 — the first derivative changes sign. At a local max, f″ < 0 (concave down); at a local min, f″ > 0 (concave up). The two concepts are distinct: an inflection point is not an extreme value of f, and a local extremum is not a change in concavity.
Why does the coefficient of x not affect the concavity of a cubic polynomial?
Concavity is determined by the second derivative f″(x). For f(x) = Ax³ + Bx² + Cx + D, the first derivative is 3Ax² + 2Bx + C, and the second derivative is 6Ax + 2B. The constant C (coefficient of x) disappears when you differentiate twice because it contributed only a linear term that vanishes. Therefore, only the coefficients A and B of the highest-degree terms influence the shape of the concavity across the curve.