91Ó°ÊÓ

In mathematics, a critical point of a differentiable function is where the function's derivative is zero or undefined. In this exercise, the function is evaluated at the point \(x = 1\) where \(f^{\prime}(1) = 0\). At a critical point, it's essential to further analyze the behavior of the derivative on either side of the point to determine whether it could be a local maximum, minimum, or possibly a point of inflection. These evaluations help sketch the graph accurately by understanding where the curve might "peak" or "valley". For instance, when you have \(f^{\prime}(1) = 0\), and changes in \(f^{\prime}(x)\) occur around \(x = 1\), you might find the point transitioning from increasing to decreasing, hinting it could be a local maximum or vice-versa for a local minimum.

Sketching the graph of a function involves understanding the function's behavior to visualize its shape. For a differentiable function like \(f(x)\) where you know the critical points, how the function increases or decreases gives insight into where the graph should "rise" or "fall". For example, if \(f^{\prime}(x) > 0\) for \(x < 1\), the function is increasing as it approaches \(x = 1\), suggesting upward movement in the graph toward the critical point. Similarly, if \(f^{\prime}(x) < 0\) past the critical point, the graph should descend after passing. Each case scenario described in the steps requires its sketch method; thus you first determine its directionality (increase or decrease), use the critical point \( (1,1) \), and then extend the graph accordingly.

Understanding when a function is increasing or decreasing over a given interval relies heavily on the derivative. If \(f^{\prime}(x) > 0\) on an interval, the function is increasing, which visually translates to the graph moving upwards. Conversely, \(f^{\prime}(x) < 0\) indicates the function is decreasing, causing the graph to slope downwards. In this exercise, you have different behaviors defined for intervals around \(x = 1\) which influence the shape:

• Case a: \(f^{\prime}(x) > 0\) for \(x < 1\) and \(f^{\prime}(x) < 0\) for \(x > 1\), illustrates an increase up to \(x = 1\) and decrease thereafter—creating a peak.
• Case b: \(f^{\prime}(x) < 0\) for \(x < 1\) and \(f^{\prime}(x) > 0\) after; this shift from decrease to increase identifies a valley (local minimum) at the point.
• Case c: \(f^{\prime}(x) > 0\) for every \(x eq 1\) suggests perpetual rise except a flat at the critical point.
• Case d: \(f^{\prime}(x) < 0\) for all \(x eq 1\) indicates the graph descends apart from the central constant point.

Local maxima and minima are specific types of critical points where the function takes on a highest or lowest value, respectively. At a local maximum, the function switches from increasing (rising) to decreasing (falling), while a local minimum marks the transition from decreasing to increasing. Determining these is crucial for plotting realistic graphs:

• Local Maximum: In our exercise, situations like where \(f^{\prime}(x) > 0\) before the point and \(f^{\prime}(x) < 0\) after the point—such as in Case a—suggest the graph reaches a top at the critical point.
• Local Minimum: This is evident in cases like Case b, where the function descends until hitting \(x = 1\) and then ascends, marking a bottom at the critical point.
• Horizontal Inflections: Sometimes changes at critical points, as in cases c or d where the function remains flat before continuing in a direction, are identified by horizontal lines through the graph instead of peaks or valleys.

By identifying these traits within a function’s derivative and behavior, you gain clarity in sketching and understanding differentiated functions.