Exercise · ZW-CRITICAL-POINTS-008 · ★★★ Advanced
How many critical points does \(f(x)=x^{2/3}\) have, and which are they?
Choose one option.
Show answer
- A. One: \(x=0\), where \(f'\) does not exist
Explanation
For \(x\neq 0\), \(f'(x)=\frac{2}{3}x^{-1/3}\) is never zero. At \(x=0\) the derivative does not exist, so it is the only critical point.
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...
- ZW-CRITICAL-POINTS-002 What are the critical points of f(x)=x^3 - 3 x?
- ZW-CRITICAL-POINTS-003 What are the critical points of f(x)=2 x^3 - 9 x^2 + 12 x?
- 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-009 If f is differentiable at c, which condition makes c a critical point?