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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 7 - Transcendental Functions문제 7.5.88c
7장, 문제 7.5.88c

88. Given that x>0, find the maximum value, if any, of
c. x^(1/x^n) (n a positive integer)

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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.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
7m
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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
가이드 코스
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.
추천 영상:
06:30
Logarithmic 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.
추천 영상:
04:50
Critical Points
관련 실천
교과서 질문

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

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교과서 질문

86. This exercise explores the difference between

lim(x→∞)(1 + 1/x²)^x

and

lim(x→∞)(1 + 1/x)^x = e

c. Confirm your estimate of lim(x→∞)f(x) by calculating it with l’Hôpital’s Rule.

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교과서 질문

10. True, or false? As x→∞,

c. 1/x - 1/x² = o(1/x)

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교과서 질문

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

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교과서 질문

4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

c. x²e^(-x)

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교과서 질문

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)

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