Calculators / Calculus 3 / Critical Points & Local Extrema Calculator

Critical Points & Local Extrema Calculator

Published on: August 2, 2026

This Critical Points & Local Extrema Calculator helps you find the critical points of a function of two variables and classify them as local maxima, local minima, or saddle points. The calculator solves ∇f = 0, then uses the Hessian test to determine the type of each critical point. This makes it easier to analyze the behavior of the function near those points. It is a simple way to check answers, understand the method clearly, and practise multivariable calculus step by step.

Step-by-step method

Identify the function f(x,y).
Compute f_x and f_y with substeps.
Solve f_x = 0 and f_y = 0 to get the critical points.
Compute f_xx, f_yy, and f_xy.
For each critical point, compute D = f_xx f_yy − (f_xy)^2.
Classify each critical point using the Hessian test.

Formulas:

Critical point condition

∇f = ⟨ f_x, f_y ⟩

Critical points satisfy: f_x = 0 and f_y = 0

Hessian determinant

D = f_xxf_yy − ( f_xy )²

Classification rules

If D > 0 and f_xx > 0, local minimum

If D > 0 and f_xx < 0, local maximum

If D < 0, saddle point

If D = 0, inconclusive

Example 1: f(x,y)=x^3-3x+y^2

Step 1 - Identify the function.

In this problem: Read the given function.

f( x, y ) = x3 + y2 − 3x

Step 2A - Differentiate term 1 with respect to x.

In this problem: Rule shown by the math.

∂

∂x

(x3) = 3x2

Step 2B - Differentiate term 2 with respect to x.

In this problem: Rule shown by the math.

∂

∂x

(y2) = 0

Step 2C - Differentiate term 3 with respect to x.

In this problem: Rule shown by the math.

∂

∂x

(−3x) = -3·

∂

∂x

(x)

Step 2D - Differentiate term 3 with respect to x.

In this problem: Rule shown by the math.

∂

∂x

(x) = 1

Step 2E - Differentiate term 3 with respect to x.

In this problem: Rule shown by the math.

∂

∂x

(−3x) = -3

Step 2F - Combine the results.

In this problem: Add the derivatives from the previous substeps.

f_x = 3x2 + 0 + -3 = 3(−1 + x2)

Step 3A - Differentiate term 1 with respect to y.

In this problem: Rule shown by the math.

∂

∂y

(x3) = 0

Step 3B - Differentiate term 2 with respect to y.

In this problem: Rule shown by the math.

∂

∂y

(y2) = 2y

Step 3C - Differentiate term 3 with respect to y.

In this problem: Rule shown by the math.

∂

∂y

(−3x) = 0

Step 3D - Combine the results.

In this problem: Add the derivatives from the previous substeps.

f_y = 0 + 2y + 0 = 2y

Step 4A - Set f_x equal to zero.

In this problem: Critical points satisfy f_x = 0.

f_x = 3(−1 + x2) = 0

Step 4B - Set f_y equal to zero.

In this problem: Critical points satisfy f_y = 0.

f_y = 2y = 0

Step 4C - Solve the system.

In this problem: Solve f_x = 0 and f_y = 0 together.

Critical points: ( -1, 0 ), ( 1, 0 )

Step 5A - Compute f_xx.

In this problem: Differentiate f_x with respect to x.

f_xx =

∂

∂x

(−3 + 3x2) = 6x

Step 5B - Compute f_yy.

In this problem: Differentiate f_y with respect to y.

f_yy =

∂

∂y

(2y) = 2

Step 5C - Compute f_xy.

In this problem: Differentiate f_x with respect to y.

f_xy =

∂

∂y

(−3 + 3x2) = 0

Step 6A - Evaluate f_xx at the critical point.

In this problem: Substitute the critical point.

f_xx( -1, 0 ) = -6

Step 6B - Evaluate f_yy at the critical point.

In this problem: Substitute the critical point.

f_yy( -1, 0 ) = 2

Step 6C - Evaluate f_xy at the critical point.

In this problem: Substitute the critical point.

f_xy( -1, 0 ) = 0

Step 6D - Compute the Hessian determinant.

In this problem: Use D = f_xx f_yy − (f_xy)^2.

D = -6·2 − ( 0 )² = -12

Step 6E - Classify the critical point.

In this problem: Use the Hessian test.

( -1, 0 ): Saddle point

Step 7A - Evaluate f_xx at the critical point.

In this problem: Substitute the critical point.

f_xx( 1, 0 ) = 6

Step 7B - Evaluate f_yy at the critical point.

In this problem: Substitute the critical point.

f_yy( 1, 0 ) = 2

Step 7C - Evaluate f_xy at the critical point.

In this problem: Substitute the critical point.

f_xy( 1, 0 ) = 0

Step 7D - Compute the Hessian determinant.

In this problem: Use D = f_xx f_yy − (f_xy)^2.

D = 6·2 − ( 0 )² = 12

Step 7E - Classify the critical point.

In this problem: Use the Hessian test.

( 1, 0 ): Local minimum

Final answer: Critical points classified

Example 2: f(x,y)=-(x^2+y^2)

Step 1 - Identify the function.

In this problem: Read the given function.

f( x, y ) = −x2 − y2

Step 2A - Differentiate term 1 with respect to x.

In this problem: Rule shown by the math.

∂

∂x

(−x2) = -1·

∂

∂x

(x2)

Step 2B - Differentiate term 1 with respect to x.

In this problem: Rule shown by the math.

∂

∂x

(x2) = 2x

Step 2C - Differentiate term 1 with respect to x.

In this problem: Rule shown by the math.

∂

∂x

(−x2) = −2x

Step 2D - Differentiate term 2 with respect to x.

In this problem: Rule shown by the math.

∂

∂x

(−y2) = 0

Step 2E - Combine the results.

In this problem: Add the derivatives from the previous substeps.

f_x = −2x + 0 = −2x

Step 3A - Differentiate term 1 with respect to y.

In this problem: Rule shown by the math.

∂

∂y

(−x2) = 0

Step 3B - Differentiate term 2 with respect to y.

In this problem: Rule shown by the math.

∂

∂y

(−y2) = -1·

∂

∂y

(y2)

Step 3C - Differentiate term 2 with respect to y.

In this problem: Rule shown by the math.

∂

∂y

(y2) = 2y

Step 3D - Differentiate term 2 with respect to y.

In this problem: Rule shown by the math.

∂

∂y

(−y2) = −2y

Step 3E - Combine the results.

In this problem: Add the derivatives from the previous substeps.

f_y = 0 + −2y = −2y

Step 4A - Set f_x equal to zero.

In this problem: Critical points satisfy f_x = 0.

f_x = −2x = 0

Step 4B - Set f_y equal to zero.

In this problem: Critical points satisfy f_y = 0.

f_y = −2y = 0

Step 4C - Solve the system.

In this problem: Solve f_x = 0 and f_y = 0 together.

Critical points: ( 0, 0 )

Step 5A - Compute f_xx.

In this problem: Differentiate f_x with respect to x.

f_xx =

∂

∂x

(−2x) = -2

Step 5B - Compute f_yy.

In this problem: Differentiate f_y with respect to y.

f_yy =

∂

∂y

(−2y) = -2

Step 5C - Compute f_xy.

In this problem: Differentiate f_x with respect to y.

f_xy =

∂

∂y

(−2x) = 0

Step 6A - Evaluate f_xx at the critical point.

In this problem: Substitute the critical point.

f_xx( 0, 0 ) = -2

Step 6B - Evaluate f_yy at the critical point.

In this problem: Substitute the critical point.

f_yy( 0, 0 ) = -2

Step 6C - Evaluate f_xy at the critical point.

In this problem: Substitute the critical point.

f_xy( 0, 0 ) = 0

Step 6D - Compute the Hessian determinant.

In this problem: Use D = f_xx f_yy − (f_xy)^2.

D = -2·-2 − ( 0 )² = 4

Step 6E - Classify the critical point.

In this problem: Use the Hessian test.

( 0, 0 ): Local maximum

Final answer: Critical points classified