Problem 69 - Thomas Calculus

Question

Locate and identify the absolute extreme values of a. \(\ln (\cos x)\) on \([-\pi / 4, \pi / 3]\) b. \(\cos (\ln x)\) on \([1 / 2,2]\)

Step-by-Step Solution

Answer

a. Max: 0 at \(x = 0\); Min: \(\ln\left(\frac{1}{2}\right)\) at \(x = \frac{\pi}{3}\). b. Max: 1 at \(x = 1\); Min: \(\cos(\ln 2)\) at \(x = \frac{1}{2}\) and \(x = 2\).

Step 1: Understand the Problem for Part a

We need to find the absolute maximum and minimum values of the function \( f(x) = \ln(\cos x) \) within the interval \( \left[-\frac{\pi}{4}, \frac{\pi}{3}\right] \). Since the natural logarithm function is involved, the function is only defined where \( \cos x > 0 \), i.e., \( x eq \pm \frac{\pi}{2} + k\pi \) for integer \( k \). In this interval, \( \cos x \) remains positive.

Step 2: Find Critical Points for Part a

Find the derivative of \( f(x) = \ln(\cos x) \), which is \( f'(x) = -\tan(x) \). Set the derivative to zero to find critical points: \( -\tan(x) = 0 \), which gives \( x = 0 \) within our interval.

Step 3: Evaluate Function at Critical Points and Endpoints for Part a

Evaluate \( f(x) = \ln(\cos x) \) at the endpoints and at the critical point within the interval:- At \( x = -\frac{\pi}{4} \): \( f\left(-\frac{\pi}{4}\right) = \ln\left(\frac{\sqrt{2}}{2}\right) \)- At \( x = 0 \): \( f(0) = \ln(1) = 0 \)- At \( x = \frac{\pi}{3} \): \( f\left(\frac{\pi}{3}\right) = \ln\left(\frac{1}{2}\right) \)

Step 4: Determine Absolute Extremes for Part a

Compare the values obtained:- \( \ln\left(\frac{\sqrt{2}}{2}\right) \approx -0.3466 \)- \( 0 \)- \( \ln\left(\frac{1}{2}\right) \approx -0.6931 \)The absolute maximum is \( 0 \) at \( x = 0 \), and the absolute minimum is \( \ln\left(\frac{1}{2}\right) \) at \( x = \frac{\pi}{3} \).

Step 5: Understand the Problem for Part b

Find the absolute maximum and minimum of \( g(x) = \cos(\ln x) \) on the interval \([\frac{1}{2}, 2]\). The natural logarithm \( \ln x \) will be real and well-defined in this interval.

Step 6: Find Critical Points for Part b

Compute the derivative \( g'(x) = -\sin(\ln x) \frac{1}{x} \). Set \( g'(x) = 0 \) to find critical points: \( -\sin(\ln x) = 0 \), which occurs when \( \ln x = n\pi \) for integer \( n \). Solve for \( x \): - \( x = e^{n\pi} \) falls outside our interval as \( e^{n\pi} > 2 \) for \( n \geq 1 \). The only relevant critical point is \( x = e^0 = 1 \) for \( n=0 \), since it falls in our interval.

Step 7: Evaluate Function at Critical Points and Endpoints for Part b

Evaluate \( g(x) = \cos(\ln x) \) at the endpoints and critical point:- At \( x = \frac{1}{2} \): \( g\left(\frac{1}{2}\right) = \cos(\ln 0.5) = \cos(-\ln 2) \)- At \( x = 1 \): \( g(1) = \cos(\ln 1) = \cos(0) = 1 \)- At \( x = 2 \): \( g(2) = \cos(\ln 2) \)

Step 8: Determine Absolute Extremes for Part b

Compare the values of \( g(x) \):- \( \cos(0) = 1 \)- \( \cos(-\ln 2) = \cos(\ln 2) \)Since \( \cos(\ln 2) \) is also positive but less than 1, the absolute maximum is \( 1 \) at \( x = 1 \). The minimum value is \( \cos(\ln 2) \) occurring at both endpoints.

Key Concepts

Critical Points

When dealing with functions and their extreme values, finding the critical points is a crucial step. Critical points are where the derivative of a function is zero or undefined, indicating potential maximum or minimum values.

• For the function \( f(x) = \ln(\cos x) \), the derivative is \( f'(x) = -\tan(x) \). Setting \( f'(x) \) to zero helps us find critical points: \( -\tan(x) = 0 \), giving \( x = 0 \) within the interval \([\frac{-\pi}{4}, \frac{\pi}{3}].\)
• For \( g(x) = \cos(\ln x) \), the derivative \( g'(x) = -\sin(\ln x) \frac{1}{x} \) is used. Setting it to zero gives \( -\sin(\ln x) = 0 \), leading to \( \ln x = n \pi \). The solution \( x = e^{n \pi} \) shows \( x = 1 \) as a critical point within \([\frac{1}{2}, 2]\).

Understanding critical points allows you to pinpoint where maxima or minima might occur within a given interval. Consider both the calculation and why these points inform about the function's behavior.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base of Euler's number \( e \approx 2.71828 \). It's widely used in calculus because it has a straightforward derivative, and \( \ln(x) \) is defined only for positive \( x \).

• In \( f(x) = \ln(\cos x) \), \( \ln(\cos x) \) is only defined where \( \cos x > 0 \). This simplifies interval analysis, constraining values within valid ranges where the natural log applies.
• For \( g(x) = \cos(\ln x) \), \( \ln x \) reinterprets \( x \) along an exponential scale, emphasizing changes in lower intervals more than higher fractions. Recognizing where \( \ln \) affects transformations helps in understanding domain limitations.

The natural logarithm's properties assist in examining growth rates and transformations, critical for interval analysis and extreme value discovery.

Interval Analysis

Interval analysis involves inspecting the behavior of a function over a specified domain, often to find extreme values. By looking through the endpoints and critical points, you can ascertain where a function peaks or bottoms out within those bounds.

• In examined functions, endpoints provide external measures, where behaviors are heightened or suppressed. For \( f(x) = \ln(\cos x) \), endpoints are \( x = \frac{-\pi}{4}, \frac{\pi}{3} \), with values reflecting \( f(x) \) across its defined interval.
• For \( g(x) = \cos(\ln x) \), endpoints \( x = \frac{1}{2}, 2 \) set \( g(x) \)'s range, bounding its value directly at these positions, illustrating changes across the interval.

Analyzing how functions operate through strategic interval points allows locating extremes efficiently, highlighting integral differences in behavior.

Trigonometric Functions

Trigonometry functions like sine and cosine showcase periodic behavior, often oscillating between fixed values. In calculus, they're essential, helping illustrate cyclical patterns and balance changes over specific intervals.

• In function \( f(x) = \ln(\cos x) \), \( \cos x \) flips through inverse cycles, influencing how \( \ln \) interprets positive outputs in the given interval.
• For \( g(x) = \cos(\ln x) \), cosine compresses \( \ln x \)'s output, rounding oscillatory peaks and troughs, checking where \( g(x) \) stabilizes or destabilizes based on \( x \).

The impeccable rhythm of trigonometric functions accounts for their recurrent applications, making them invaluable in extreme value computations that underscore periodic movements.