Question

In Exercises \(1-6,\) use the First Derivative Test to determine the local extreme values of the function, and identify any absolute extrema. Support your answers graphically. $$y=\left\\{\begin{array}{l}{3-x^{2}} \\ {x^{2}+1}\end{array}\right.\( \)x<0\( \)x \geq 0$$

Short answer

To solve this problem, take derivatives of each function, apply the first derivative test to identify local extrema, and observe the overall behavior of the function to find any absolute extrema. Additionally, provide a graph for visual support.

Step-by-step solution

Identify Different Functions

Firstly, identify the two separate functions within the piecewise function. They are \(3-x^{2}\) for \(x<0\) and \(x^{2}+1\) for \(x \geq 0\). The two functions are split at \(x = 0\), so take special note of the values of the functions at \(x = 0\).

Compute the Derivatives

Then, compute the derivatives of each piece of the piecewise function: \(-2x\) for \(3-x^{2}\) and \(2x\) for \(x^{2}+1\). Set each derivative equal to zero and solve for \(x\) to find critical points.

Apply the First Derivative Test

Apply the first derivative test to find minimum or maximum points. If a function changes from negative to positive at an \(x\) value, then that \(x\) value is a local minimum. If a function changes from positive to negative at an \(x\) value, then that \(x\) value is a local maximum.

Identify Absolute Extrema

Identify the absolute extrema (if any) by considering the behaviour of the function at the critical points and at the extreme values of its domain. In this case, the domain of the function is all real numbers, so check the behavior of the function as \(x\) approaches infinity and negative infinity.

Support Your Answers Graphically

Lastly, draw a graph to visually represent the function and its extrema. Plot the critical points and trace the function. The extreme points will be apparent visually on the graph.