Chapter 3: Q. 84 (page 311)

Now that we know L’Hopital’s rule, we can apply it to solve more sophisticated global optimization problems. Consider domains, limits, derivatives, and values to determine the global extrema of each function $f$ in Exercises 81–86 on the given intervals $I$ and $J$ .
$f (x) = \frac{\ln x}{\ln 2 x}, I = [0, 1], J = [1, \infty)$ .

Short Answer

Expert verified

On $I, f$ has neither a global maximum nor a global minimum.

On $J, f$ has neither a global maximum nor a global minimum.

Step by step solution

Step 1. Given information

$f (x) = \frac{\ln x}{\ln 2 x}, I = [0, 1], J = [1, \infty)$ .

Step 2. Now, for critical point, f'x=0.

$\frac{d}{d x} (\frac{\ln x}{\ln 2 x}) = 0 \frac{\ln (2 x) \cdot \frac{1}{x} - \ln x \cdot \frac{1}{2 x} \cdot 2}{(\ln 2 x)^{2}} = 0 \frac{1}{x} \ln (2 x) - \frac{1}{x} \ln x = 0 \ln 2 x - \ln x = 0 \ln \frac{2 x}{x} = 0 \ln 2 = 0$

This is a false statement. Therefore, the function has no critical point.

Step 3. Again,

$\lim_{x \to 0} f (x) = \lim_{x \to 0} \frac{\ln x}{\ln 2 x}$ [ in the form of $\frac{\infty}{\infty}$ ]

$= \lim_{x \to 0} \frac{\frac{1}{x}}{\frac{1}{2 x} \cdot 2}$ [Using L'Hopital's rule]

$= \lim_{x \to 0} \frac{\frac{1}{x}}{\frac{1}{x}} = \lim_{x \to 0} 1 = 1 \lim_{x \to 1} f (x) = \lim_{x \to 1} \frac{\ln x}{\ln 2 x} = \frac{\ln 1}{\ln 2} = 0$

Step 4. The graph of the function with limit I=0,1 is shown below:

Step 5.Therefore, on I,f has neither a global maximum nor a global minimum.

Again,

$\lim_{x \to \infty} f (x) = \lim_{x \to \infty} \frac{\ln x}{\ln 2 x}$ [ in the form of $\frac{\infty}{\infty}$ ]

$= \lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{1}{2 x} \cdot 2}$ [ Using L'Hopital's rule]

$= \lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{1}{x}} = \lim_{x \to \infty} 1 = 1$

Step 6. The graph for the function with limit J=1,∞ is shown below:

Therefore, on $J, f$ has neither a global maximum nor a global minimum.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Now, for critical point, f'x=0.

Step 3. Again,

Step 4. The graph of the function with limit I=0,1 is shown below:

Step 5.Therefore, on I,f has neither a global maximum nor a global minimum.

Step 6. The graph for the function with limit J=1,∞ is shown below:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Calculus

Geometry

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Now that we know L’Hopital’s rule, we can apply it to solve more sophisticated global optimization problems. Consider domains, limits, derivatives, and values to determine the global extrema of each function fin Exercises 81–86 on the given intervals Iand J.f(x)=lnxln2x,I=[0,1],J=[1,∞).

Short Answer

Step by step solution

Step 1. Given information

Step 2. Now, for critical point, f'x=0.

Step 3. Again,

Step 4. The graph of the function with limit I=0,1 is shown below:

Step 5.Therefore, on I,f has neither a global maximum nor a global minimum.

Step 6. The graph for the function with limit J=1,∞ is shown below:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Calculus

Geometry

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.