Textbook Question
1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
c. √x
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.5.88c
88. Given that x>0, find the maximum value, if any, of
c. x^(1/x^n) (n a positive integer)
Verified step by step guidance1
Start by expressing the given function in a form that is easier to differentiate. The function is \(f(x) = x^{\frac{1}{x^n}}\). To simplify differentiation, take the natural logarithm: \(\ln f(x) = \frac{1}{x^n} \ln x\).
Define \(g(x) = \ln f(x) = \frac{\ln x}{x^n}\). To find critical points of \(f(x)\), find where the derivative of \(g(x)\) equals zero, since \(f(x)\) and \(g(x)\) have maxima at the same \(x\) values (because the logarithm is a monotonic function).
Differentiate \(g(x)\) using the quotient rule: \(g'(x) = \frac{(1/x) \cdot x^n - \ln x \cdot n x^{n-1}}{(x^n)^2}\). Simplify the numerator and denominator carefully.
Set \(g'(x) = 0\) and solve for \(x\). This will give the critical points where the function \(f(x)\) could have maxima or minima. Remember, \(x > 0\) and \(n\) is a positive integer.
Analyze the critical points by using the second derivative test or by examining the behavior of \(f(x)\) around these points to determine if the critical points correspond to a maximum value.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Analysis and Domain
Understanding the domain of the function x^(1/x^n) is crucial, especially since x > 0 and n is a positive integer. This ensures the function is well-defined and real-valued, allowing us to analyze its behavior and find extrema within the positive real numbers.
Recommended video:
5:10
Finding the Domain and Range of a Graph
Logarithmic Differentiation
Logarithmic differentiation simplifies finding the derivative of functions with variable exponents, like x^(1/x^n). By taking the natural logarithm, the exponent becomes a product, making it easier to apply differentiation rules and identify critical points.
Recommended video:
06:30Logarithmic Differentiation
Finding and Classifying Critical Points
To find the maximum value, we set the derivative equal to zero to locate critical points. Then, using the second derivative test or analyzing the sign changes of the first derivative, we determine whether these points correspond to maxima, minima, or neither.
Recommended video:
04:50Critical Points
Related Practice
Textbook Question
c. Find the slopes of the tangent lines to the graphs of f and g at (1, 1) and (−1, −1) (four tangent lines in all).
Textbook Question
10. True, or false? As x→∞,
c. 1/x - 1/x² = o(1/x)
Textbook Question
In Exercises 41–44:
c. Evaluate df/dx at x = a and df⁻¹/dx at x = f(a) to show that
(df⁻¹/dx)|ₓ₌f(a) = 1 / (df/dx)|ₓ₌a
44. f(x) = 2x², x ≥ 0, a = 5
Textbook Question
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
Textbook Question
2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
c. √(1+x^4)
1
views