Find extrema
Set fx(x,y) and fy(x,y) to 0, plug in boundaries of x and y into original function.
To find the extrema of a function, you need to find its critical points. A critical point is a point where the derivative of the function is zero or does not exist. Once you have identified the critical points, you can then use the second derivative test to determine whether each critical point is a maximum, minimum or neither.
Here is the step-by-step process to find the extrema:
1. Find the derivative of the function using the power rule, product rule, quotient rule, or chain rule as needed.
2. Find all the critical points by setting the derivative equal to zero and solving for x. If the derivative does not exist at a point, it could also be a critical point.
3. Locate the endpoints of the domain of the function if it is a closed interval.
4. Use the second derivative test to identify the type of each critical point. To use this test, evaluate the second derivative at each critical point. If the second derivative is positive, the point is a minimum; if it is negative, the point is a maximum; if it is zero, the test is inconclusive.
5. Determine whether the endpoints of the domain can be extrema by evaluating the function at these points and comparing the values.
6. Write a conclusion that summarizes the location and type of all the extrema.
It is important to note that this process applies to finding extrema for a univariate function. For multivariate functions, the process involves finding partial derivatives and critical points by setting them equal to zero.
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