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Local extrema refer to points in a function's domain where the function takes on a maximum or minimum value in a small neighborhood around those points. In multivariable calculus, local extrema can be thought of as the peaks (local maxima) and valleys (local minima) of a surface, while saddle points are areas where the function does not have a direct peak or valley but instead resembles a saddle.

To find local extrema, you need to analyze the regions on the function’s graph where these local maxima, minima, or saddle points occur. Visual inspectin – through graphs or level curves – helps in giving an initial estimate of where these points might be. However, actual verification requires calculus techniques, particularly the use of partial derivatives, to pinpoint these locations precisely.

In the given function, we are primarily interested in finding these extreme points within the specified domain defined by the intervals for x and y.

Partial derivatives help us understand how a function changes as each variable is varied independently, while keeping the other variable constant. In the context of multivariable calculus functions like our example, partial derivatives are crucial for exploring changes in the surface.

The partial derivative with respect to x, denoted as \( \frac{\partial f}{\partial x} \), gives us the rate of change of the function as x changes. Similarly, the partial derivative with respect to y, \( \frac{\partial f}{\partial y} \), focuses on changes along the y direction.

In our specific problem,

• For \( \frac{\partial f}{\partial x} = \cos x - \sin(x+y) \), we understand how the function reacts horizontally.
• For \( \frac{\partial f}{\partial y} = \cos y - \sin(x+y) \), we observe vertical behavior.

Setting these partial derivatives to zero is the key step to finding critical points, as it helps identify where the function stops increasing or decreasing - potential spots for local extrema.

The second partial derivative test is a method used to classify critical points found using the first derivatives. This test helps determine whether these critical points are local minima, local maxima, or saddle points.

For a function \( f(x, y) \), the second partial derivatives involved are:

• \( \frac{\partial^2 f}{\partial x^2} \) - the curvature in the direction of x
• \( \frac{\partial^2 f}{\partial y^2} \) - the curvature in the direction of y
• \( \frac{\partial^2 f}{\partial x \partial y} \) - the mixed derivative indicating how x and y interact

Using these, we calculate the discriminant, \( D = \frac{\partial^2 f}{\partial x^2} \cdot \frac{\partial^2 f}{\partial y^2} - \left( \frac{\partial^2 f}{\partial x \partial y} \right)^2 \). The sign and conditions of D tell us:

• If \( D > 0 \) and \( \frac{\partial^2 f}{\partial x^2} < 0 \), we have a local maximum.
• If \( D > 0 \) and \( \frac{\partial^2 f}{\partial x^2} > 0 \), it's a local minimum.
• If \( D < 0 \), it indicates a saddle point.

In our solution, this test helps confirm whether the point \( (\pi/4, \pi/4) \) is indeed a local extremum.

Critical points are the locations on a surface where the partial derivatives are zero. This means there is no rate of change in any direction from the point. Identifying these points is crucial for finding where the function might achieve local maxima, minima, or neither.

The process involves:

• Calculating the partial derivatives of the function.
• Setting these derivatives equal to zero to form equations.
• Solving these equations simultaneously to find specific values of x and y, which are the critical points.

For the function \( f(x, y) = \sin x + \sin y + \cos (x+y) \), we identified and solved:

• \( \cos x = \sin(x+y) \)
• \( \cos y = \sin(x+y) \)

By finding solutions such as \( x = y = \pi/4 \), we ascertain the potential location of local extrema within the given domain, which then undergo further testing for confirmation of their nature.