Optimization Analysis with Exponential-Log Functions

00:00 5 marks

A chemical engineer models the efficiency of a reaction using the function \(f(x) = x + e^{-x} – \ln(x)\), where \(x\) (in arbitrary units) represents a temperature-related parameter and \(x > 0\). For experimental purposes, the parameter is restricted to the interval \([0.5, 3]\).

Determine the stationary point of \(f(x)\) within the interval and decide whether it is a local maximum or local minimum. Then, by evaluating \(f(x)\) at the stationary point and at the endpoints, identify the global minimum of the function on \([0.5, 3]\).

Hint Hint

Hint

First differentiate \(f(x)\) to find its stationary point by setting \(f'(x)=0\). Then calculate \(f(x)\) at x = 0.5, the stationary point, and x = 3 to compare the values.

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Common Errors Common Errors
Real World Context Real World Context
Focus Focus
Formula Booklet

Answer Answer

Complete Answer

The function has a unique stationary point at approximately \(x \approx 1.35\), which is a local minimum. Evaluating, \(f(0.5) \approx 1.80\), \(f(1.35) \approx 1.31\), and \(f(3) \approx 1.95\); hence the global minimum on \([0.5, 3]\) is at \(x \approx 1.35\) with \(f(1.35) \approx 1.31\).

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Common Errors Common Errors

Focus Focus

Formula Booklet