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A continuous function is one that can be drawn without lifting your pen from the paper. This simply means, at every point of the graph, the function has no breaks, jumps, or holes. The importance of continuity in graph sketching is paramount because it dictates how the function behaves along its range. For the function \(g(x)\) given in the exercise, it's mentioned to be continuous at all points and specifically at \(x = 2\). This implies not just existing values for every input near and at \(x = 2\), but also that its overall behavior doesn't disrupt around this point. Even where the derivative changes signs, continuity ensures smooth transition from one portion of the curve to the next, seamlessly linking every segment of the graph.

The derivative of a function, \(g'(x)\), is crucial for understanding the slope of the graph at any given point, which represents the rate of change at that point. In this exercise, the behavior of the derivative around \(x = 2\) gives us specifics that assist in predicting the shape of the function.

• Part (a) specifies a positive derivative approaching 1 as \(x \to 2^-\), indicating an upward slope that gets less steep as it nears \(x = 2\). Conversely, the negative derivative \(g'(x)\) moving towards 0 after \(x = 2\) hints at a downward slope that flattens out.
• In Part (b), the derivative becomes strongly negative approaching \(-\infty\) before reaching \(x = 2\), and drastically shifts to positive trending towards \(+\infty\) after 2, suggesting a dramatic change in steepness on either side of \(x = 2\).

This derivative behavior is indicative of how rapidly the function shifts from one direction to another at \(x = 2\).

Critical points in a function are where the derivative either vanishes (equals zero) or becomes undefined. They are typically where the function's graph has local maxima, minima, or point of inflection. For \(g(x)\), since \(g(2) = 2\), we have an explicit point on the graph at \((2, 2)\). In part (a), this is interpreted as a local maximum because the graph slopes positively until \(x = 2\), then switches downward. In part (b), the behavior around this point indicates a cusp, caused by the sudden and large shift in derivative values from \(-\infty\) to \(+\infty\), so while not a traditional critical point by the derivative being zero, it's critical by the behavior shift.

Concavity refers to the direction in which a graph "bends." If a function is concave up, it is bent upwards like a cup and its second derivative is positive. Conversely, concave down indicates it bends downwards and has a negative second derivative. In this problem, concavity helps further illustrate the changes in behavior of the graph.

• In Part (a), the function's gradual flattening after \(x=2\) as \(g'(x)\) approaches zero suggests a transition from concave up to concave down around \(x=2\), forming a slight, smooth peak.
• Part (b) shows a stark transition at \(x=2\) with very sharp changes in slope that leave a cusp, displaying how rapid changes in gradient can manifest visually as points of infinite concavity.

Understanding concavity is essential in sketching graphs as it gives further insight into the deeper geometry of the graph and whether bends create peaks, valleys, or cusps as in this exercise.