Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. then f must have a critical point at (a, b).

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.  then f must have a critical point at (a, b).

Quizplus · Accepted Answer

This question is referring to the multivariable calculus concept of the gradient vector, which is a vector that points in the direction of greatest increase of a function in space. If f has a critical point at (a, b), that means that the gradient of f at that point is equal to the zero vector. In other words, all partial derivatives of f evaluated at (a, b) are equal to zero. By the multivariable calculus theorem known as the "gradient test", if f has a critical point at (a, b), then the gradient vector of f at that point must be the zero vector. Conversely, if the gradient vector of f at a point (a, b) is the zero vector, then (a, b) is a critical point of f. Therefore, the statement is true: if 11eaa8b1_d9c7_f720_b1b6_df9ae1906f01_TB6026_11, then f must have a critical point at (a, b).

Determine Whether the Statement Is True or False