Applications of derivatives · Definition

Critical points

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Before you start you need to know

A critical point is where the derivative \(f'\) vanishes or does not exist. These points are the candidates for local extrema and guide the study of how \(f\) grows.

Definition

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.

\[f'(c)=0\ \ \text{or}\ \ f'(c)\ \text{does not exist}\]

A critical point is a value of \(x\) in the domain where \(f'(x)=0\) or \(f'\) does not exist. They are the only candidates for interior local maxima and minima.

Careful: being critical does not guarantee being an extremum. For \(x^3\), \(x=0\) is critical and yet the function keeps increasing.

The procedure is: differentiate, find where it vanishes or fails, then classify using the sign of the derivative or the second derivative.

Think about it

Is \(x=0\) an extremum of \(x^3\)?

No: \(f'(0)=0\) but \(f'\) does not change sign; it is critical, not an extremum.

Exercises 10

  1. A critical point of f is a number c in the domain where...
  2. What are the critical points of \(f(x)=x^{3} - 3 x\)?
  3. What are the critical points of \(f(x)=2 x^{3} - 9 x^{2} + 12 x\)?
  4. What are the critical points of \(f(x)=x^{4} - 4 x^{3}\)?
  5. What are the critical points of \(f(x)=x^{3} - 12 x + 5\)?
  6. What value of \(x\) is a critical point of \(f(x)=x^{2}-6x\)?
  7. What are the critical points of \(f(x)=x^{4}-8x^{2}\)?
  8. How many critical points does \(f(x)=x^{2/3}\) have, and which are they?
  9. If \(f\) is differentiable at \(c\), which condition makes \(c\) a critical point?
  10. What is the critical point of \(f(x)=x e^{x}\)?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.