Test on Derivative of a function

Question 1: If f′(c) = 0, the Second Derivative Test will always definitively determine whether f(c) is a local maximum or a local minimum.

A. Yes

B. No

Explanation: The study materials state that if f″(c) = 0, the Second Derivative Test is inconclusive, meaning it cannot definitively determine if f(c) is a local maximum or minimum in that specific case.

Question 2: According to the provided study materials, what is the *first* step mentioned for analyzing the polynomial function f(x) = x³ − 6x² + 9x + 1?

A. Determine the intervals where the function is concave up or down.

B. Find the second derivative of the function.

C. Find the first derivative of the function.

D. Identify any potential inflection points by setting f″(x) = 0.

Explanation: The 'Worked Example: Analyzing a Polynomial Function' section explicitly states, '1. Find the first derivative: Using the power and sum/difference rules:' as the initial step for analyzing the given function f(x) = x³ − 6x² + 9x + 1.

Question 3: Is it a common error in calculus to assume that the derivative of a product is the product of the individual derivatives?

A. Yes

B. No

Explanation: The study materials highlight that a common mistake is to assume the derivative of a product is the product of the derivatives, or similarly for quotients.

Question 4: Based on the provided analysis of the function f(x) = x³ − 6x² + 9x + 1, which of the following statements correctly identifies the local extrema?

A. There is a local maximum at (1, 5) and a local minimum at (3, 1).

B. There is a local minimum at (1, 5) and a local maximum at (3, 1).

C. There is a local maximum at (2, 3) and a local minimum at (3, 1).

D. The function has no local extrema.

Explanation: The study material states: 'At x = 1, f′(x) changes from positive to negative, so there is a local maximum. f(1) = 5. Local maximum at (1, 5).' It also states: 'At x = 3, f′(x) changes from negative to positive, so there is a local minimum. f(3) = 1. Local minimum at (3, 1).'

Question 5: In an interactive simulation, an inflection point is characterized by the first derivative graph crossing the x-axis.

A. Yes

B. No

Explanation: The study materials state that an inflection point is where the second derivative graph will cross the x-axis, not the first derivative graph. The first derivative indicates local extrema when it crosses or touches the x-axis (is zero), not inflection points.