Find the interval on which the function
𝑓
(
𝑥
)
=
𝑥
ln
𝑥
is increasing.

Question

Find the interval on which the function 
𝑓
(
𝑥
)
=
𝑥
ln
𝑥
 is increasing.

BlackTom AI · Accepted Answer

The problem asks for the interval where f(x) = x ln x is increasing.
First, note the function's domain: x > 0, since ln x is defined only for positive x.
Next, compute the derivative: f'(x) = d/dx [x ln x] = ln x + 1, using the product rule.
To determine where the function is increasing, we solve f'(x) > 0, i.e., ln x + 1 > 0.
This inequality is equivalent to ln x > -1, which translates (by exponentiating both sides) to x > e^{-1} = 1/e.
Hence, f is increasing on the interval (1/e, ∞).
Now consider the provided option: (
1
/
𝑒
,
𝑖nfty
)
, which corresponds to (1/e, ∞). This matches our calculated interval exactly.
If there were other options, they would need to satisfy the same inequality f'(x) > 0 with x > 0, i.e., x > 1/e, or else be invalid due to domain constraints. Any interval including x ≤ 1/e would fail the increasing condition at points where f'(x) ≤ 0, and intervals extending into x ≤ 0 would be outside the domain entirely.
Therefore, the given option correctly identifies the increasing interval for f(x) = x ln x.

MATH-112-301-001 Unproctored Midcourse Exam 3 Practice Exam 2

Find the interval on which the function 𝑓 ( 𝑥 ) = 𝑥 ln 𝑥 is increasing.

View Explanation

Log in for full answers

Similar Questions

Given a function whose graph is shown below, under which interval/s is the graph increasing? A. ( − 3 , − 1 ) ∪ ( 2 , 4 ) B. ( − 1 , + ∞ ) C. ( − 3 , 1 ) D. ( − ∞ , − 3 ) ∪ ( − 1 , 2 ) ∪ ( 4 , + ∞ )

State when the function is increasing and decreasing using interval notation. Use inf for ∞\infty. Increasing: [Fill in the blank], Decreasing: [Fill in the blank],

State when the function is increasing and decreasing using interval notation. Use inf for ∞\infty. Increasing: [Fill in the blank], Decreasing: [Fill in the blank],

Let . Select all answers that correctly finish the following sentence: is increasing...

Let g be the function given by 𝑔 ( 𝑥 ) = ∫ 0 𝑥 3 sin ⁡ ( 𝑡 2 ) 𝑑 𝑡 for − 1 ≤ 𝑥 ≤ 3 . Function g(x) is DECREASING on which interval?

The rate of change of temperature of hot-air ballon is given by r(t)=t3−5t2+9 for 0≤t≤8. What is the change in the temperature of the ballon during the interval the temperature is DECREASING?

Suppose a function f is continuous on [a,b], differentiable on (a,b), and f′(x)≠0 for all a<x<b. Then f attains both of its absolute extrema at the endpoints a and b.

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler