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In the realm of mathematics, piecewise functions are functions composed of multiple sub-functions, each having its own specific interval of the domain. They offer a way to define a function in pieces, with different behaviors in different sections. In our specific example, the function is defined as:

• \(f(x) = 3 - x^2\) for \(x < 0\)
• \(f(x) = x^2 + 1\) for \(x \geq 0\)

Each piece is governed by its own rule or equation. Depending on the interval of \(x\), you will use the corresponding equation to evaluate the function. This kind of function is particularly useful in situations where a variable experiences abrupt changes under different conditions. When working with piecewise functions, understanding these intervals and their corresponding equations is crucial for accurate calculations and interpretations.

Local extrema refer to the minimum and maximum points within a particular range or interval of a function. The First Derivative Test is commonly used to find these points. For our piecewise function, we need to determine where the function's slope changes direction, as this indicates a potential local extremum.To identify such points:

• Find the derivative of each piece of the function.
• Set the derivative equal to zero and solve for \(x\) to find critical points.

For \(f(x) = 3 - x^2\), the derivative is \(-2x\). The derivative becomes zero at \(x = 0\), but since our interval is \(x < 0\), there are no critical points here.For \(f(x) = x^2 + 1\), the derivative is \(2x\), and this also equates to zero at \(x = 0\), positioning it as a critical point for the interval \(x \geq 0\).Understanding local extrema helps in graphically summarizing the increasing or decreasing nature of the function and pinpointing specific locations of peak behaviour or troughs on the graph.

Critical points are crucial for understanding where a function might achieve a maximum, minimum, or neither. They occur where the derivative of the function is either zero or undefined.For our piecewise function:

• Derivative of \(3 - x^2\) is \(-2x\). No zero points since interval is \(x < 0\).
• Derivative of \(x^2 + 1\) is \(2x\). Zero point exists where \(x = 0\) for \(x \geq 0\).

After determining where the derivative is zero for each piece, we further use the First Derivative Test to analyze the changes in the slope to classify these points as either maxima or minima. In our case, \(x = 0\) for the interval \(x \geq 0\) is a critical point where the slope direction might change.

Graphical analysis is a valuable tool in understanding functions visually by interpreting their shape and behavior. It aids in confirming analytical solutions by providing a visual walkthrough.Here's how to proceed with our piecewise function:

• Plot the graph separately for both \(f(x) = 3 - x^2\) (a downward-facing parabola) and \(f(x) = x^2 + 1\) (an upward-facing parabola).
• Each piece should be plotted only in its defined domain.
• Mark the critical point at \(x = 0\) for the interval \(x \geq 0\) found analytically.

Understand that at \(x=0\), the transition between the pieces happens. Verify local extrema by observing direction changes in the graph. The visual plot will confirm that for \(f(x) = x^2 + 1\), \(x=0\) appears as a local minimum. Using graphical analysis alongside derivative findings presents a holistic view of the function's behavior.