Sketching F(x) Using Derivatives Finding Y'' And Graphing Procedure

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Determining the Second Derivative: y''

To find the second derivative, we need to differentiate the first derivative, y' = x(x-18)^2, with respect to x. Let's break down the process:

  1. Expand the first derivative: y' = x(x^2 - 36x + 324) = x^3 - 36x^2 + 324x

  2. Differentiate term by term: y'' = d/dx (x^3) - d/dx (36x^2) + d/dx (324x)

  3. Apply the power rule: y'' = 3x^2 - 72x + 324

  4. Factor out a common factor (optional): y'' = 3(x^2 - 24x + 108)

  5. Factor the quadratic expression (optional): y'' = 3(x - 6)(x - 18)

Thus, the second derivative of the function is y'' = 3(x - 6)(x - 18). The second derivative, y'', provides insights into the concavity of the function's graph. It tells us where the graph is concave up (opening upwards like a cup) and concave down (opening downwards like a frown).

Graphing Procedure: Unveiling the Shape of f(x)

Now that we have both the first and second derivatives, we can employ the graphing procedure to sketch the general shape of the graph of f(x). This procedure involves several key steps:

  1. Find the critical points:

    • Critical points are the points where the first derivative, y', is equal to zero or undefined. These points are potential locations of local maxima, local minima, or saddle points.
    • Set y' = x(x - 18)^2 = 0 and solve for x.
    • The solutions are x = 0 and x = 18. These are our critical points.

  2. Determine the intervals of increasing and decreasing:

    • The first derivative, y', tells us where the function is increasing or decreasing.
    • If y' > 0, the function is increasing.
    • If y' < 0, the function is decreasing.
    • Create a sign chart for y' using the critical points as dividing points.
    • Test values in each interval to determine the sign of y':
      • For x < 0, y' is negative, so the function is decreasing.
      • For 0 < x < 18, y' is positive, so the function is increasing.
      • For x > 18, y' is positive, so the function is increasing.

  3. Find the inflection points:

    • Inflection points are the points where the concavity of the graph changes. These points occur where the second derivative, y'', is equal to zero or undefined.
    • Set y'' = 3(x - 6)(x - 18) = 0 and solve for x.
    • The solutions are x = 6 and x = 18. These are potential inflection points.

  4. Determine the intervals of concavity:

    • The second derivative, y'', tells us about the concavity of the graph.
    • If y'' > 0, the graph is concave up.
    • If y'' < 0, the graph is concave down.
    • Create a sign chart for y'' using the potential inflection points as dividing points.
    • Test values in each interval to determine the sign of y'':
      • For x < 6, y'' is positive, so the graph is concave up.
      • For 6 < x < 18, y'' is negative, so the graph is concave down.
      • For x > 18, y'' is positive, so the graph is concave up.

  5. Analyze critical points and inflection points:

    • Critical points:
      • At x = 0, the function changes from decreasing to increasing, indicating a local minimum.
      • At x = 18, the first derivative is zero, but it does not change sign, indicating a saddle point (a point where the function flattens out but does not change direction).
    • Inflection points:
      • At x = 6, the concavity changes from up to down, confirming an inflection point.
      • At x = 18, the concavity changes from down to up, and we already identified this as a saddle point. So, it's also an inflection point.

  6. Sketch the graph:

    • Plot the critical points and inflection points.
    • Use the information about increasing/decreasing intervals and concavity to sketch the shape of the graph in each interval.
    • Remember that the function is continuous, so the graph should be smooth and connected.

General Shape of the Graph of f(x)

Based on our analysis, the general shape of the graph of f(x) can be described as follows:

  • The graph has a local minimum at x = 0.
  • The graph has a saddle point and an inflection point at x = 18.
  • The graph is decreasing for x < 0.
  • The graph is increasing for x > 0.
  • The graph is concave up for x < 6.
  • The graph is concave down for 6 < x < 18.
  • The graph is concave up for x > 18.

By combining this information, we can sketch a general shape of the graph of f(x). The graph will have a valley at x = 0, a flattened area at x = 18, and changes in concavity at x = 6 and x = 18.

Conclusion

By finding the first and second derivatives of the function y = f(x) and employing the graphing procedure, we can gain a comprehensive understanding of the function's behavior and sketch its general shape. The derivatives provide valuable information about critical points, intervals of increasing and decreasing, concavity, and inflection points, allowing us to create an accurate representation of the function's graph. This process highlights the power of calculus in analyzing and visualizing functions.

This detailed analysis provides a roadmap for understanding the behavior of the function f(x). Remember that this is a general shape, and the specific vertical position and scaling of the graph would require additional information or the original function f(x) itself.