Hessian determinant calculator

It then applies the second derivative test when the point is critical, distinguishing a local minimum, local maximum, saddle point, or inconclusive case. Every derivative is obtained analytically from the polynomial rather than estimated from nearby samples, making the output suitable for checking calculus exercises, preparing teaching material, and adding predictable mathematical validation to an application.

Enter the function and point in a supported form

Write the function as an expanded polynomial in x and y, then provide finite numeric values for both coordinates. Inputs such as x^2 + 3xy + 2y^2, 4*x^3*y - 7xy + 5, and f(x,y)=x^2-y^2 are accepted. Multiplication signs between ordinary factors are optional, and either ^ or ** can introduce a nonnegative integer exponent. Decimal coefficients and scientific notation are also supported. The parser deliberately rejects parentheses, division by variables, negative or fractional powers, symbolic coefficients, and named functions such as sine, logarithm, or exponential. Keeping the accepted grammar explicit prevents an unsupported expression from being interpreted approximately or only in part. Like terms are combined internally, so x^2 + 2x^2 is treated as 3x^2. The coordinate fields must be actual finite numbers, not numeric strings, infinity, or missing values. Limits on expression length, number of terms, and exponent size keep every request bounded. If your function is not a polynomial, simplify it only when doing so is mathematically exact; otherwise use a tool designed for that function family.

Understand the Hessian determinant calculation

For a function f(x,y), the calculator first evaluates the gradient components fx and fy. It then differentiates again to obtain fxx, fxy, fyx, and fyy. For a polynomial, the mixed partial derivatives agree, so fxy and fyx are returned with the same value. These four entries form the two-by-two Hessian matrix. Its determinant is D=fxx times fyy minus the square of fxy. The implementation derives every coefficient and power term by term, evaluates the resulting derivatives at the supplied coordinates, and normalizes insignificant floating-point residue for stable output. It does not estimate slopes through a step size, so there is no finite-difference interval to choose and no truncation error caused by sampling neighboring points. The response includes the function value, gradient, Hessian entries, and determinant, allowing you to audit the calculation instead of receiving only a classification label. You can verify the determinant directly by multiplying the returned diagonal entries and subtracting fxy multiplied by fyx. All arithmetic must remain finite; requests whose magnitudes overflow ordinary number arithmetic are rejected rather than returning an unusable infinity or null value.

Apply and interpret the second derivative test

The second derivative test classifies a point only after the gradient is zero. The calculator therefore returns a critical flag and reports not_critical_point when either first partial derivative is nonzero, even if the Hessian determinant itself happens to be positive or negative. At a critical point, a positive determinant with positive fxx indicates a local minimum, while a positive determinant with negative fxx indicates a local maximum. A negative determinant indicates a saddle point because the surface curves in opposite directions. When the determinant is zero, the test is inconclusive: the point may still be a minimum, maximum, saddle, or none of these, but higher-order analysis is required. This distinction matters because zero is not evidence that nothing happens. Results use concise machine-readable labels so they can be tested in educational software or processing pipelines, while all contributing numbers remain visible for human review. Browser calculations are free, and automated API requests use the published base price of $0.002. Since execution uses no network, randomness, saved state, or clock, the same valid input produces the same structured result across supported channels.