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The first step in tackling any calculus problem involving increasing or decreasing functions is derivative analysis. This involves finding the derivative of the given function. The derivative, denoted as \( f'(x) \), gives us critical information about the slope of the function at any point. For the function \( f(x) = 5 + 12x - x^3 \), we calculate the first derivative to explore these changes:

• Function: \( f(x) = 5 + 12x - x^3 \)
• First Derivative: \( f'(x) = 12 - 3x^2 \)

By understanding what the derivative tells us, students can determine whether the function \( f(x) \) is increasing, decreasing, or remaining constant over particular intervals. This forms the foundation needed to move on to analyzing other features of the graph, such as critical points and concavity.

Critical points are key values of \( x \) where the derivative \( f'(x) \) equals zero or is undefined. At these points, the function's behavior changes, indicating potential maximums, minimums, or points of inflection. By setting \( f'(x) = 12 - 3x^2 \) to zero, we find:

• Critical points occur at \( x = \pm 2 \)

After identifying the critical points, students can also determine on which intervals the function is increasing or decreasing:- Test intervals around \( x = -2 \) and \( x = 2 \): - \( f'(x) < 0 \) when \( x < -2 \) and \( x > 2 \) (decreasing intervals). - \( f'(x) > 0 \) between \( x = -2 \) and \( x = 2 \) (increasing interval).Understanding critical points helps students predict the behavior of the function across its domain and analyze the nature of these points.

Concavity involves understanding the curvature of a function, which is determined via the second derivative. The second derivative, denoted \( f''(x) \), helps determine whether a function is concave up or concave down over certain intervals. For \( f(x) = 5 + 12x - x^3 \), the second derivative is:

• Second derivative: \( f''(x) = -6x \)

By assessing the sign of \( f''(x) \), we determine concavity:- If \( f''(x) > 0 \), the curve is concave up.- If \( f''(x) < 0 \), the curve is concave down.Testing around potential inflection points, we understand:- The function is concave up when \( x < 0 \) and concave down when \( x > 0 \).These insights allow students to visualize how the function curves across different values of \( x \) and inform the overall shape of the graph.

An inflection point is where the function changes its concavity, switching from concave up to concave down, or vice versa. This change is particularly noticeable where \( x \)-values make the second derivative go from positive to negative or vice versa. For the function \( f(x) = 5 + 12x - x^3 \), solve \( f''(x) = -6x = 0 \) to find:

• Inflection point: \( x = 0 \)

At this point, concavity shifts:- From concave up on \( (-\infty, 0) \) to concave down on \( (0, \infty) \).Inflection points provide critical insight into graphing the function, ensuring students understand where curves shift direction. Evaluating \( f(x) \) at the inflection point \( x = 0 \), we find its coordinate is \( (0, 5) \). This leads to a fuller analysis and presentation of the function's key features.