How to Find Maxima and Minima in Two Variables

TL;DR

Find stationary points by setting both first partial derivatives equal to zero, then classify each point using the second partial derivatives. If the determinant formed from the second derivatives is positive, the sign of the repeated x derivative distinguishes a minimum from a maximum; a negative determinant gives neither, while a zero determinant leaves the test inconclusive.

Transcript

Hello students, I Dr. Gajendra Purohit and today we are going to discuss maxima and minima of two variables so let us see what it is this is a bit advanced then what we discussed in 12 th class and if any variable is in two dimension we will apply partial differentiation first of all, we will see the condition in any question first, we will calc... Read More

Key Insights

• A stationary point is found by calculating the first partial derivatives with respect to both variables, setting each derivative equal to zero, and solving the two equations together for the variable values.
• The second derivative test uses r, s, and t, where r and t are repeated partial derivatives and s is a mixed partial derivative. These quantities must be evaluated at each stationary point.
• The classification determinant is obtained by multiplying r and t and subtracting the square of s. Its sign determines whether further classification as a maximum or minimum is possible.
• A positive classification determinant with r greater than zero identifies a local minimum. The minimum function value is then calculated by substituting the stationary point into the original function.
• A positive classification determinant with r less than zero identifies a local maximum. The maximum function value is obtained by evaluating the original function at that classified stationary point.
• A negative classification determinant means the stationary point is neither a maximum nor a minimum. The lecture demonstrates that different stationary points of one function can produce different determinant signs.
• A zero classification determinant is a doubtful or inconclusive case under this test. The trigonometric example includes such a point, which is left unclassified before another stationary point is checked.
• Every stationary point must be tested separately because solving the first derivative equations can produce several points. One example produces four points, including a maximum, a minimum, and two points classified as neither.

Questions & Answers

Q: How do you find stationary points of a two-variable function?

Calculate the partial derivative of the function with respect to the first variable and set it equal to zero. Then calculate the partial derivative with respect to the second variable and also set it equal to zero. Solve the two equations together. The resulting coordinate pairs are stationary points that must be tested individually for maximum or minimum behavior.

Q: How are maxima and minima classified with second partial derivatives?

After finding a stationary point, calculate the second partial derivatives denoted in the lecture by r, s, and t. Form the classification value by multiplying r and t and subtracting the square of s. If that value is positive, use the sign of r to distinguish a minimum from a maximum. Other determinant signs require different conclusions.

Q: When is a stationary point a local minimum?

A stationary point is classified as a minimum when the second derivative classification value is greater than zero and r, the repeated partial derivative with respect to the first variable, is also greater than zero. After this classification, substitute the stationary point coordinates into the original function to obtain the corresponding minimum value.

Q: When is a stationary point a local maximum?

A stationary point is classified as a maximum when the second derivative classification value is greater than zero while r, the repeated partial derivative with respect to the first variable, is less than zero. The maximum value is then found by placing the coordinates of that point into the original two-variable function and simplifying the result.

Q: What does a negative second derivative determinant mean?

A negative value of the classification expression means the stationary point is neither a maximum nor a minimum. The lecture applies this rule to multiple stationary points and shows that some points can fail to be extrema even when both first partial derivatives vanish there. Each point therefore requires its own second derivative calculation.

Q: What happens when the second derivative test gives zero?

When the classification expression equals zero, the test is described as doubtful or inconclusive because it cannot determine whether the point is a maximum, a minimum, or neither. In the trigonometric example, the zero-result point is left unresolved with this method, and another stationary point is evaluated using the same second derivative procedure.

Q: Why must every stationary point be checked separately?

The equations formed by setting the first partial derivatives equal to zero can produce several coordinate pairs. Their classifications may differ because r, s, t, and the resulting determinant can take different values at different points. One worked example has four points, with one maximum, one minimum, and two points that are neither maxima nor minima.

Q: How is the maximum or minimum function value calculated?

First classify the stationary point using the second partial derivatives and their determinant. Once the point is confirmed as a maximum or minimum, substitute its variable values into the original function. Simplify that expression to obtain the maximum or minimum value. Classification and value calculation are separate steps, so the point must be tested before evaluation.

Summary & Key Takeaways

• The procedure begins by partially differentiating the function with respect to each variable and setting both resulting expressions equal to zero. Solving these equations produces stationary points where the function may change behavior. Every resulting point must then be examined because different points from the same function can receive different classifications.

• For classification, the lecture denotes the second derivatives by r, s, and t, then evaluates the determinant formed by multiplying r and t and subtracting the square of s. A positive determinant indicates a maximum or minimum, while a negative determinant identifies a point that is neither.

• Several worked examples demonstrate how to solve for stationary points, substitute their coordinates into the second derivatives, classify them, and calculate the corresponding function values. The examples include variables written as x and y or x1 and x2, multiple stationary points, and a trigonometric case where a zero determinant is inconclusive.