Question

The minimum value of $x\log x$ is equal to

• A. $e$
• B. $1/e$
• C. $-1/e$
• D. $2/e$

Hint:

$\dfrac{d}{dx}(x\ln x) = \ln x + 1$. The minimum of $x\ln x$ occurs at $x = 1/e$ and equals $-1/e$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Find the critical point of $f(x) = x\ln x$ for $x>0$ and classify it.
Step 2: Detailed Explanation:
$f'(x) = \ln x + 1 = 0 \Rightarrow x = 1/e$.
$f''(x) = 1/x>0$, so it is a minimum.
$f(1/e) = (1/e)\ln(1/e) = (1/e)(-1) = -1/e$.
Step 3: Final Answer:
Minimum value $= -1/e$.