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Chapter 2: Problem 45

In Exercises, display the graph of the derivative of \(f(x)\) in the specified window. Then use the graph of \(f^{\prime}(x)\) to determine the approximate values of \(x\) at which the graph of \(f(x)\) has relative extreme points and inflection points. Then check your conclusions by displaying the graph of \(f(x)\). $$f(x)=\left(x^{2}\right) 3 x^{5}-20 x^{3}-120 x ;[-4,4] \text { by }[-325,325]$$

Short Answer

Expert verified
Approximate the relative extreme and inflection points by analyzing the graph of the derivative. Verify by graphing the original function.

Step by step solution

01

- Derive the Function

Compute the derivative of the function. Given: \[ f(x) = 3x^7 - 20x^5 - 120x \] Apply the power rule: \[ f'(x) = \frac{d}{dx} (3x^7) - \frac{d}{dx} (20x^5) - \frac{d}{dx} (120x) = 21x^6 - 100x^4 - 120 \]
02

- Display the Graph of the Derivative

Graph \( f'(x) = 21x^6 - 100x^4 - 120 \) within the window \([-4,4] \text{ by } [-325,325] \) using graphing technology.
03

- Determine Relative Extreme Points

Find where the graph of \( f'(x) \) crosses the x-axis. These are the critical points where \( f(x) \) might have relative maxima or minima. Approximate these points using the graph.
04

- Determine Inflection Points

Identify where the derivative changes concavity, which is where the second derivative \( f''(x) \) changes sign. These are the inflection points. Use the graph to approximate these points.
05

- Check Your Conclusions

Graph the original function \( f(x) = 3x^7 - 20x^5 - 120x \) in the same window \([-4,4] \text{ by } [-325,325] \) to verify the relative extreme points and inflection points identified from \( f'(x) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivatives
In calculus, a derivative represents how a function changes as its input changes. Essentially, it provides the slope of the function at any given point. This is a fundamental tool because it helps us understand rates of change and motion. For a function like the one given in the exercise,
  • The function is given by f(x) = 3x^7 - 20x^5 - 120x
  • Taking the derivative means applying the power rule, a basic technique. Each term in the function is derived individually.
Applying the power rule example: The derivative of x^n is nx^(n-1). Therefore, f'(x) = 21x^6 - 100x^4 - 120
  • This equation represents the slope of each point in f(x).
    • Derivatives are crucial because they allow us to locate critical points, assess the function's behavior, and identify relative extreme points and inflection points.
    Relative Extreme Points
    Relative extreme points are the points at which a function reaches a local maximum or minimum. These are essential in graph analysis because they help in understanding the function鈥檚 behavior and turning points. To find these points:
    • First, locate where the derivative f'(x) crosses the x-axis. This means finding the roots of the equation 21x^6 - 100x^4 - 120=0.
    These points are where the slope is zero, meaning the function changes direction. At these critical points, f(x) could have peaks (local maxima) or troughs (local minima). To confirm whether these points are indeed maxima or minima, we can:
    • Use the second derivative test: If the second derivative, f''(x), is positive at a critical point, f(x) has a local minimum there. If it鈥檚 negative, f(x) has a local maximum.
    Finding and understanding these relative extreme points helps in sketching and understanding the overall shape of the function graph.
    Inflection Points
    Inflection points are points on the graph of a function at which the concavity changes. In simpler terms, these are points where the graph changes its curvature direction (from concave up to concave down or vice versa). To locate inflection points:
    • Derive the second derivative of the function. For our function, f(x), the second derivative is obtained by differentiating f'(x).
    To find where f''(x) changes signs, solve the equation f''(x)=0. However, find the points by checking where the sign of f''(x) actually changes. These points illustrate where the slope of the slope (the rate of change of the rate of change) varies, marking key points in the graph where the curvature flips.
    Power Rule
    The power rule is a fundamental rule in calculus for finding the derivative of functions in the form of x raised to a power. It鈥檚 one of the first rules you'll learn because of its simplicity and wide application.

    • The general form of the power rule is: If f(x) = x^n, then f'(x) = nx^(n-1).
    For example, applying it to our function:
    • For the term 3x^7, the derivative is 21x^6.
    • For -20x^5, it becomes -100x^4.
    • And, for -120x, it鈥檚 simply -120 (since x to the power of 1 diminishes to 1-1=0).

    Using the power rule simplifies differentiation, making it easier to find the slope and analyze the graph of functions. Recognizing how to apply this rule efficiently is vital for more complex calculus problems and graph analysis.

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