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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.1.35
Capitolo 4, Problema 4.1.35

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.


ƒ(x) = 1 / x + ln x

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To find the critical points of the function \( f(x) = \frac{1}{x} + \ln x \), we first need to find its derivative \( f'(x) \).
Differentiate \( f(x) \) with respect to \( x \). The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \) and the derivative of \( \ln x \) is \( \frac{1}{x} \). Therefore, \( f'(x) = -\frac{1}{x^2} + \frac{1}{x} \).
Set the derivative \( f'(x) \) equal to zero to find the critical points: \( -\frac{1}{x^2} + \frac{1}{x} = 0 \).
Solve the equation \( -\frac{1}{x^2} + \frac{1}{x} = 0 \) for \( x \). This involves finding a common denominator and simplifying the equation.
After solving the equation, determine the values of \( x \) that satisfy it. These values are the critical points of the function \( f(x) \).

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Critical Points

Critical points of a function occur where its derivative is either zero or undefined. These points are essential for identifying local maxima, minima, and points of inflection. To find critical points, one must first compute the derivative of the function and then solve for the values of x that satisfy the condition of the derivative being zero or undefined.
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Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus that provides information about the function's slope at any given point. For the function ƒ(x) = 1/x + ln x, the derivative must be calculated to locate the critical points, which involves applying the rules of differentiation.
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Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a key function in calculus, particularly in integration and differentiation. In the context of the given function, understanding the properties of the natural logarithm is crucial for correctly differentiating and analyzing the function's behavior near its critical points.
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