Math Problem Statement

To find the absolute minimum value of the function f(x)=xln⁡x, we need to follow these steps:

Solution

To find the absolute minimum value of the function f(x)=xlnxf(x) = x \ln x, follow these steps:

  1. Domain: Identify the domain of the function. The function lnx\ln x is defined for x>0x > 0, so the domain of f(x)f(x) is (0,)(0, \infty).

  2. Critical Points: Find the critical points by taking the derivative and setting it equal to zero. f(x)=xlnxf(x) = x \ln x f(x)=lnx+1f'(x) = \ln x + 1 Set the derivative equal to zero: lnx+1=0\ln x + 1 = 0 lnx=1\ln x = -1 x=e1=1ex = e^{-1} = \frac{1}{e}

  3. Second Derivative Test: Use the second derivative to determine whether this critical point is a minimum or maximum. f(x)=1xf''(x) = \frac{1}{x} At x=1ex = \frac{1}{e}: f(1e)=e>0f''\left(\frac{1}{e}\right) = e > 0 Since the second derivative is positive, x=1ex = \frac{1}{e} is a local minimum.

  4. Value at Critical Point: Evaluate the function at the critical point to find the minimum value. f(1e)=1eln(1e)f\left(\frac{1}{e}\right) = \frac{1}{e} \ln \left(\frac{1}{e}\right) f(1e)=1e(1)=1ef\left(\frac{1}{e}\right) = \frac{1}{e} \cdot (-1) = -\frac{1}{e}

  5. Boundary Behavior: Check the behavior of f(x)f(x) as x0+x \to 0^+ and xx \to \infty.

    • As x0+x \to 0^+: limx0+xlnx=0\lim_{x \to 0^+} x \ln x = 0 because lnx\ln x \to -\infty and xlnxx \ln x approaches 0.
    • As xx \to \infty: limxxlnx\lim_{x \to \infty} x \ln x \to \infty because lnx\ln x grows slower than xx, making xlnxx \ln x grow to \infty.

Given the analysis, the absolute minimum value of the function f(x)=xlnxf(x) = x \ln x is 1e-\frac{1}{e}, which occurs at x=1ex = \frac{1}{e}.

Would you like more details on any step or have any questions?

Here are some related questions to consider:

  1. What is the domain of the natural logarithm function?
  2. How do you find critical points of a function?
  3. What is the significance of the second derivative test?
  4. What happens to f(x)f(x) as xx approaches 0 from the right?
  5. Why does the function xlnxx \ln x approach \infty as xx increases?
  6. What are some common techniques to determine the behavior of a function at infinity?
  7. How does the behavior of lnx\ln x differ from polynomial and exponential functions?
  8. What are the differences between local and absolute extrema of a function?

Tip: Always check the domain and boundary behavior of the function to understand its overall behavior.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Critical Points
Second Derivative Test
Limit Analysis

Formulas

Derivative formula
Second derivative test formula

Theorems

Second derivative test

Suitable Grade Level

Undergraduate