CalculusLesson 1 of 1 in the course Applications of derivatives

Critical Points Across Functions, Curves, and Smooth Maps

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

This lesson introduces critical points and critical values for functions of one real variable. These ideas help identify places where a function may have a local maximum or minimum.

For a real-valued function ff, a critical point is an input x0x_0 in the domain of ff where either f′(x0)=0f'(x_0)=0 or ff is not differentiable at x0x_0. When the function is differentiable at a critical point and its derivative is zero, the point is also called a stationary point.

All flashcards

Critical point of a single-variable function
An input in the domain where the derivative is zero or the function is not differentiable.
Critical value
The function output at a critical point.
Stationary point
A differentiable critical point where the derivative is zero.
Hessian matrix
The matrix of second partial derivatives of a multivariable function.
Regular value
A value that is not the image of any critical point.

Quiz questions

  1. At a smooth point of F(x,y)=0F(x,y)=0, what equations locate critical points for projection onto the xx-axis?Answer: F(x,y)=0F(x,y)=0 and Fy(x,y)=0F_y(x,y)=0.
  2. At a nondegenerate critical point, what does a Hessian with both positive and negative eigenvalues indicate?Answer: The point is a saddle point.
  3. For a differentiable map, what is a regular value?Answer: A value that is not the image of a critical point.
  4. What does Sard's theorem state about the critical values of a smooth map?Answer: Their set has measure zero.
  5. At a nondegenerate critical point, how is the Hessian index defined?Answer: As the number of negative Hessian eigenvalues.

Based on the Wikipedia article Critical point (mathematics), licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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