Exercise · ZW-CRITICAL-POINTS-010 · ★★☆ Intermediate
What is the critical point of \(f(x)=x e^{x}\)?
Choose one option.
Show answer
- C. \(x=-1\)
Explanation
By the product rule, \(f'(x)=(1+x)e^{x}\). Since \(e^{x}\neq 0\), we need \(1+x=0\), so \(x=-1\).
Review the concept: Critical points
A critical point of \(f\) is a number \(c\) in the domain where \(f^{\prime}(c)=0\) or \(f^{\prime}(c)\) does not exist. Local maxima and minima of a function at an interior point occur at critical points.
\[f'(c)=0\ \ \text{or}\ \ f'(c)\ \text{does not exist}\]
More practice problems: critical points
- ZW-CRITICAL-POINTS-001 A critical point of f is a number c in the domain where...
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- ZW-CRITICAL-POINTS-004 What are the critical points of f(x)=x^4 - 4 x^3?
- ZW-CRITICAL-POINTS-005 What are the critical points of f(x)=x^3 - 12 x + 5?
- ZW-CRITICAL-POINTS-006 What value of x is a critical point of f(x)=x^2-6x?
- ZW-CRITICAL-POINTS-007 What are the critical points of f(x)=x^4-8x^2?
- ZW-CRITICAL-POINTS-008 How many critical points does f(x)=x^2/3 have, and which are they?