Test on Local extrema and inflection points

Question 1: A local extremum can only occur at a point where the first derivative of the function is equal to zero.

A. Yes

B. No

Explanation: The study materials state that 'Critical points also include points where the derivative is undefined', implying that a local extremum can exist at a point where the derivative is undefined, not just where it is zero. Points where the derivative is zero are specifically called 'stationary points', which are candidates for local extrema, but not the only type of critical point where they can occur.

Question 2: The function f(x) = x³ - 3x² + 2 has a local maximum at f(0) = 2.

A. Yes

B. No

Explanation: According to the 'Worked Example' section, for the function f(x) = x³ - 3x² + 2, the first derivative test shows that f′(x) changes from positive to negative at x = 0, indicating that f(0) = 2 is a local maximum.

Question 3: Which of the following are steps in the comprehensive function analysis procedure as described in the study materials for curve sketching?

A. Calculating the first derivative to find stationary points.

B. Finding potential inflection points by setting the second derivative to zero.

C. Identifying the domain and intercepts of the function.

D. Using the limit definition of the derivative to confirm stationary points.

Explanation: The study materials list 'First Derivative Analysis' which includes calculating f'(x) and finding stationary points by solving f'(x) = 0. 'Second Derivative Analysis' includes calculating f''(x) and finding potential inflection points by solving f''(x) = 0. 'Domain and Intercepts' is listed as the first step. The limit definition of the derivative is not mentioned as a step in the comprehensive function analysis procedure.

Question 4: According to the study materials, which of the following statements about a stationary point of a function f(x) is true?

A. The second derivative of the function at that point must be positive or negative, but not zero.

B. The function's first derivative at that point is zero.

C. The function must always have either a local maximum or a local minimum at that point.

D. The function's graph changes concavity at that point.

Explanation: The study materials define stationary points as points where 'f′(x) = 0'. While stationary points are candidates for local extrema, the materials explicitly state that 'not all stationary points are local extrema; some can be inflection points'. Therefore, a local extremum is not guaranteed, nor is a change in concavity, which is characteristic of an inflection point. The second derivative can be zero at a stationary point (in which case the Second Derivative Test is inconclusive), so the first option is incorrect.

Question 5: A simulation to visualize local extrema primarily involves inputting trigonometric functions to observe their derivative graphs.

A. Yes

B. No

Explanation: The study materials indicate that a simulation to visualize local extrema allows users to input simple polynomial functions such as quadratic, cubic, or quartic functions and observe how their graphs change. There is no mention of primarily inputting trigonometric functions or observing derivative graphs for this specific purpose in the provided text.