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Chapter 3: Q. 81 (page 311)

determine the global extrema of function $f (x) = x \ln (x), I = (0, 1), J = (0, 鈭�)$

Short Answer

Expert verified

on I, f has a global maximum at x=1 and a global minimum at $x = \frac{1}{e}$

on J, f has no global maximum and a global minimum at $x = \frac{1}{e}$

Step by step solution

01

Step 1. Given information

The given information is $f (x) = x \ln (x), I = (0, 1), J = (0, 鈭�)$

02

Step 2. Calculate the function

$f (x) = x \ln (x) f' (x) = 0 \frac{d (x \ln (x))}{d x} = 0 x 路 \frac{1}{x} + \ln (x 路 1) = 0 \ln (x) = - 1 x = \frac{1}{e}$

similarly

$\lim_{x 鈫� 0} f (x) = \lim_{x 鈫� 0} x \ln (x) = \lim_{x 鈫� 0} \frac{\ln (x)}{\frac{1}{x}} = \lim_{x 鈫� 0} \frac{\frac{1}{x}}{- \frac{1}{x^{2}}} = \lim_{x 鈫� 0} \frac{1}{x} (- x^{2}) = \lim_{x 鈫� 0} (- x) = 0$

Therefore on I, f has a global maximum at x=1 and a global minimum at $x = \frac{1}{e}$

Again

$\lim_{x 鈫� 鈭�} f (x) = \lim_{x 鈫� 鈭�} x \ln (x) = 鈭�$

Therefore on J, f has no global maximum, and a global minimum at $x = \frac{1}{e}$ because $\lim_{x 鈫� 鈭�} f (x) = 鈭�$

03

Step 3. The solution

Therefore,

on I, f has a global maximum at x=1 and a global minimum at $x = \frac{1}{e}$

On J, f has no global maximum and a global minimum at $x = \frac{1}{e}$

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Most popular questions from this chapter

Sketch careful, labeled graphs of each function f in Exercises 63鈥�82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f, f', a n d f'',$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f (x) = \frac{e^{x}}{x}$

For the graph of f in the given figure, approximate all the values x 鈭� (0, 4) for which the derivative of f is zero or does not exist. Indicate whether f has a local maximum, minimum, or neither at each of these critical points.

Find the relationship between between the functions $g (x) + 10 and f (x)$ .

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical point. $f (x) = x^{3} + x^{2} + 1$

Explain the difference between two antiderivatives of the function.

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