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.

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