The absolute minimum value of the function f(x)=10x(2−ln x) on the interval [1,e^2]is

To find the absolute minimum value of the function \( f(x) = 10x(2 - \ln x) \) on the interval \([1, e^2]\), we follow these steps:

### Step 1: Find the derivative of \( f(x) \)
We need to differentiate the function \( f(x) = 10x(2 - \ln x) \).

First, apply the product rule:
\[
f'(x) = \frac{d}{dx} [10x] \cdot (2 - \ln x) + 10x \cdot \frac{d}{dx} (2 - \ln x)
\]

\[
f'(x) = 10(2 - \ln x) + 10x \cdot (-\frac{1}{x})
\]

\[
f'(x) = 10(2 - \ln x) - 10
\]

\[
f'(x) = 20 - 10\ln x - 10
\]

\[
f'(x) = 10(1 - \ln x)
\]

### Step 2: Find the critical points
Set the derivative equal to zero to find the critical points:
\[
10(1 - \ln x) = 0
\]

\[
1 - \ln x = 0
\]

\[
\ln x = 1
\]

\[
x = e
\]

### Step 3: Evaluate the function at the critical points and endpoints
We now evaluate \( f(x) \) at the critical point \( x = e \) and at the endpoints of the interval \([1, e^2]\).

1. **At \( x = 1 \):**
   \[
   f(1) = 10 \cdot 1 \cdot (2 - \ln 1) = 10 \cdot 1 \cdot (2 - 0) = 20
   \]

2. **At \( x = e \):**
   \[
   f(e) = 10 \cdot e \cdot (2 - \ln e) = 10 \cdot e \cdot (2 - 1) = 10e
   \]

   Since \( e \approx 2.718 \), \( 10e \approx 27.18 \).

3. **At \( x = e^2 \):**
   \[
   f(e^2) = 10 \cdot e^2 \cdot (2 - \ln e^2) = 10 \cdot e^2 \cdot (2 - 2) = 0
   \]

### Step 4: Determine the absolute minimum
Among the values computed:
- \( f(1) = 20 \)
- \( f(e) \approx 27.18 \)
- \( f(e^2) = 0 \)

The absolute minimum value of \( f(x) \) on the interval \([1, e^2]\) is **0** at \( x = e^2 \).

Discover how to find the absolute minimum value of the function f(x) = 10x(2 - ln x) on the interval [1, e^2]. Follow a step-by-step solution involving calculus and optimization techniques. Includes evaluations at critical points and endpoints. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation