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Chapter 5: Problem 73

Find any relative extrema of the function. \(f(x)=\arctan x-\arctan (x-4)\)

Short Answer

Expert verified
After finding the first and second derivatives and critical points, test these values in the second derivative to determine if they are a relative minimum or maximum. The final solution will be the critical points and whether they are a relative minimum or maximum.

Step by step solution

01

- Calculating the Derivatives

First, calculate the first derivative of the function using the chain rule for differentiation. If \(f(x) = \arctan u \), \(f'(x) = \frac{u'}{1 + u^2}\). \nFor \(f(x)=\arctan x-\arctan (x-4)\), the first derivative \(f'(x) = \frac{1}{1 + x^2} - \frac{1}{1 + (x-4)^2}\). To find the second derivative, apply the chain rule again obtaining \(f''(x)\).
02

- Finding Critical Points

Set the first derivative \(f'(x) = 0\) in order to find the potential locations of relative extrema. Therefore, solve the equation \(\frac{1}{1 + x^2} - \frac{1}{1 + (x-4)^2}= 0\). The solutions of this equation are the critical points.
03

- Determine type of Extrema

Substitute the critical points into \(f''(x)\). If \(f''(x)>0\), it is a relative minimum. If \(f''(x)<0\), it is a relative maximum.

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Key Concepts

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

Relative extrema
In calculus, relative extrema refer to the relative maximum or minimum of a function within a certain interval. These are the points on the graph where the function rises to a peak (a maximum) or falls to a valley (a minimum) compared to its immediate surroundings.
Relative extrema are particularly useful when analyzing the behavior of functions since they indicate where the function changes direction.
  • A relative maximum occurs when a function changes from increasing to decreasing.
  • A relative minimum occurs when the function changes from decreasing to increasing.
Finding these points involves deriving the function and testing certain values to determine their nature. The first and second derivatives play a key part in identifying these extrema.
Chain rule
The chain rule is a crucial technique in calculus for finding the derivative of composite functions. A composite function is essentially one function nested within another, like peeling layers from an onion.
The chain rule states that if a variable, say, y depends on t through another variable x (where y is a function of x, and x is a function of t), then the rate of change of y concerning t can be found using the relationship:
  • First find the derivative of the outer function with respect to the inner function.
  • Then, multiply this by the derivative of the inner function with respect to the variable.
For example, for a function like \(f(x) = \arctan u\), one can find its derivative using the rule \(f'(x) = \frac{u'}{1 + u^2}\). This approach is essential for tackling derivatives of more complex functions.
First derivative
The first derivative of a function, often represented as \(f'(x)\), gives us information about the rate of change or slope of the function at any given point.
In simpler terms, it tells us how the function is behaving:
  • If \(f'(x) > 0\), the function is increasing at x.
  • If \(f'(x) < 0\), the function is decreasing at x.
  • If \(f'(x) = 0\), there might be a turning point, signaling a potential relative extremum.
Calculating the first derivative is the initial step toward finding critical points, which are essential in determining the function's behavior and identifying relative extrema locations.
Critical points
Critical points are the x-values where the first derivative of the function is zero or undefined. Investigating these points is essential to discovering where potential relative extrema might reside.
To comprehend critical points, consider the following steps:
  • Find \(f'(x)\) and set it to zero, i.e., solve \(f'(x) = 0\).
  • Alternatively, identify where \(f'(x)\) is undefined.
  • The solutions to these conditions give the x-values of the critical points.
At these x-values, the function either reaches a peak, valley, or maintains a constant slope. Further analysis with the second derivative can help determine whether these critical points are indeed maxima, minima, or even saddle points.

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Most popular questions from this chapter

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