Answer The Following Questions About The Function Whose Derivative Is

Absolutely! Here's a comprehensive article on analyzing a function based on its derivative:

Unlocking Secrets: Analyzing a Function from its Derivative

The derivative of a function, often denoted as f'(x), is a powerful tool that unveils a wealth of information about the original function, f(x). By carefully examining the derivative, we can determine where the original function is increasing or decreasing, identify local extrema (maximum and minimum points), and even gain insights into the concavity and inflection points of the function. This process of "reading" the derivative is fundamental in calculus and has wide-ranging applications in physics, economics, engineering, and many other fields.

What Can the Derivative Tell Us?

The derivative f'(x) tells us the instantaneous rate of change of the function f(x) at any given point. Here's a breakdown of the key information we can extract:

• Increasing/Decreasing Behavior:

  • If f'(x) > 0 on an interval, then f(x) is increasing on that interval.
  • If f'(x) < 0 on an interval, then f(x) is decreasing on that interval.
  • If f'(x) = 0 on an interval, then f(x) is constant on that interval.

• Critical Points: These are the points where f'(x) = 0 or f'(x) is undefined. Critical points are potential locations of local maxima, local minima, or points where the function changes direction.

• Local Extrema:

  • A local maximum occurs at a critical point c if f'(x) changes from positive to negative at c.
  • A local minimum occurs at a critical point c if f'(x) changes from negative to positive at c.

• Concavity:

  • If f''(x) > 0 on an interval, then f(x) is concave up (shaped like a "U") on that interval.
  • If f''(x) < 0 on an interval, then f(x) is concave down (shaped like an upside-down "U") on that interval.

• Inflection Points: These are the points where the concavity of f(x) changes. Inflection points occur where f''(x) = 0 or f''(x) is undefined.

Steps to Analyze a Function from its Derivative

Let's outline a step-by-step approach to analyze a function using its derivative:

1. Find the Derivative: If you're not given the derivative, f'(x), you'll need to calculate it using the rules of differentiation (power rule, product rule, quotient rule, chain rule, etc.).

2. Find Critical Points: Set f'(x) = 0 and solve for x. Also, identify any values of x where f'(x) is undefined. These are your critical points.

3. Create a Sign Chart for f'(x):

   • Draw a number line.
   • Mark all critical points on the number line.
   • Choose test values in each interval created by the critical points.
   • Evaluate f'(x) at each test value.
   • Determine the sign (+ or -) of f'(x) in each interval.

4. Determine Intervals of Increasing and Decreasing:

   • If f'(x) > 0 on an interval, f(x) is increasing on that interval.
   • If f'(x) < 0 on an interval, f(x) is decreasing on that interval.

5. Identify Local Extrema:

   • If f'(x) changes from positive to negative at a critical point, there's a local maximum at that point.
   • If f'(x) changes from negative to positive at a critical point, there's a local minimum at that point.

6. Find the Second Derivative (if needed): Calculate f''(x), the derivative of f'(x).

7. Find Potential Inflection Points: Set f''(x) = 0 and solve for x. Also, identify any values of x where f''(x) is undefined.

8. Create a Sign Chart for f''(x):

   • Draw a number line.
   • Mark all potential inflection points on the number line.
   • Choose test values in each interval created by the potential inflection points.
   • Evaluate f''(x) at each test value.
   • Determine the sign (+ or -) of f''(x) in each interval.

9. Determine Intervals of Concavity:

   • If f''(x) > 0 on an interval, f(x) is concave up on that interval.
   • If f''(x) < 0 on an interval, f(x) is concave down on that interval.

10. Identify Inflection Points:

    • If f''(x) changes sign at a potential inflection point, then it's a true inflection point.

11. Sketch the Graph (optional): Use the information gathered to sketch a rough graph of f(x). Plot the critical points, local extrema, and inflection points. Indicate the intervals of increasing/decreasing and concavity.

Illustrative Examples

Let's work through a couple of examples to solidify the process:

Example 1:

Suppose we have the derivative f'(x) = 3x² - 12. Let's analyze the original function f(x).

1. Critical Points:

   • Set f'(x) = 0: 3x² - 12 = 0
   • Solve for x: 3x² = 12 => x² = 4 => x = ±2

2. Sign Chart for f'(x):

   • Number line with -2 and 2 marked.

   Interval	Test Value	f'(x) = 3x² - 12	Sign	Increasing/Decreasing
   x < -2	x = -3	3(-3)² - 12 = 15	+	Increasing
   -2 < x < 2	x = 0	3(0)² - 12 = -12	-	Decreasing
   x > 2	x = 3	3(3)² - 12 = 15	+	Increasing

3. Increasing/Decreasing Intervals:

   • f(x) is increasing on (-∞, -2) and (2, ∞).
   • f(x) is decreasing on (-2, 2).

4. Local Extrema:

   • At x = -2, f'(x) changes from + to -, so there's a local maximum at x = -2.
   • At x = 2, f'(x) changes from - to +, so there's a local minimum at x = 2.

5. Second Derivative:

   • f''(x) = d/dx (3x² - 12) = 6x

6. Potential Inflection Points:

   • Set f''(x) = 0: 6x = 0 => x = 0

7. Sign Chart for f''(x):

   • Number line with 0 marked.

   Interval	Test Value	f''(x) = 6x	Sign	Concavity
   x < 0	x = -1	6(-1) = -6	-	Concave Down
   x > 0	x = 1	6(1) = 6	+	Concave Up

8. Concavity Intervals:

   • f(x) is concave down on (-∞, 0).
   • f(x) is concave up on (0, ∞).

9. Inflection Point:

   • At x = 0, f''(x) changes sign, so there's an inflection point at x = 0.

Example 2:

Let f'(x) = (x - 1) / x².

1. Critical Points:

   • Set f'(x) = 0: (x - 1) / x² = 0 => x - 1 = 0 => x = 1
   • f'(x) is undefined at x = 0.

2. Sign Chart for f'(x):

   • Number line with 0 and 1 marked.

   Interval	Test Value	f'(x) = (x - 1) / x²	Sign	Increasing/Decreasing
   x < 0	x = -1	(-1 - 1) / (-1)² = -2	-	Decreasing
   0 < x < 1	x = 0.5	(0.5 - 1) / (0.5)² = -2	-	Decreasing
   x > 1	x = 2	(2 - 1) / 2² = 0.25	+	Increasing

3. Increasing/Decreasing Intervals:

   • f(x) is decreasing on (-∞, 0) and (0, 1).
   • f(x) is increasing on (1, ∞).

4. Local Extrema:

   • At x = 1, f'(x) changes from - to +, so there's a local minimum at x = 1.
   • At x = 0, there is no local extrema because f(x) is not defined.

5. Second Derivative:

   • f''(x) = d/dx ((x - 1) / x²) = d/dx (x⁻¹ - x⁻²) = -x⁻² + 2x⁻³ = (-x + 2) / x³

6. Potential Inflection Points:

   • Set f''(x) = 0: (-x + 2) / x³ = 0 => -x + 2 = 0 => x = 2
   • f''(x) is undefined at x = 0.

7. Sign Chart for f''(x):

   • Number line with 0 and 2 marked.

   Interval	Test Value	f''(x) = (-x + 2) / x³	Sign	Concavity
   x < 0	x = -1	(-(-1) + 2) / (-1)³ = -3	-	Concave Down
   0 < x < 2	x = 1	(-1 + 2) / 1³ = 1	+	Concave Up
   x > 2	x = 3	(-3 + 2) / 3³ = -1/27	-	Concave Down

8. Concavity Intervals:

   • f(x) is concave down on (-∞, 0) and (2, ∞).
   • f(x) is concave up on (0, 2).

9. Inflection Points:

   • At x = 2, f''(x) changes sign, so there's an inflection point at x = 2.
   • At x = 0, there is no inflection point because f(x) is not defined.

Common Pitfalls and Considerations

• Discontinuities: The analysis above assumes that f(x) and f'(x) are continuous except at a finite number of points. If there are discontinuities, you need to analyze the behavior of the function around those points separately.
• Endpoints: If you're analyzing a function on a closed interval [a, b], you also need to consider the endpoints a and b. These can be locations of absolute (global) maxima or minima.
• Undefined Derivatives: Remember to consider points where f'(x) or f''(x) are undefined. These can be critical points or potential inflection points.
• The Constant of Integration: When working with the derivative, you only get information about the shape of the original function. To find the exact function, you need to integrate f'(x) and determine the constant of integration using additional information, such as a point on the original function.

Why is this Important?

The ability to analyze a function from its derivative is a cornerstone of calculus and has profound applications across various fields:

• Optimization: Finding the maximum or minimum value of a function is a common problem in many disciplines. For example, a business might want to maximize profit, or an engineer might want to minimize the cost of a structure.
• Curve Sketching: Understanding the shape of a function is crucial for visualization and problem-solving.
• Related Rates: Many real-world problems involve rates of change that are related to each other. By using derivatives, we can analyze and solve these problems.
• Physics: Derivatives are used to describe velocity, acceleration, and other important physical quantities.
• Economics: Derivatives are used to analyze marginal cost, marginal revenue, and other economic concepts.

Conclusion

Analyzing a function based on its derivative is a fundamental skill in calculus. By understanding the relationship between a function and its derivative, we can gain valuable insights into the behavior of the function, including where it's increasing or decreasing, its local extrema, and its concavity. This knowledge is essential for solving a wide range of problems in mathematics, science, and engineering. By mastering this technique, you'll unlock a powerful tool for understanding and analyzing the world around you.