Increasing/decreasing functions; f'(x) graphically practice questions.

Let f(x) = x³ − 3x. Determine the interval on which f is decreasing.

• x > 1
• −1 < x < 1
• x < −1
• f is never decreasing

Answer: −1 < x < 1

f'(x) = 3x² − 3 = 3(x − 1)(x + 1), which is negative exactly when −1 < x < 1, so f decreases there and increases outside this interval.

A function f satisfies f'(x) = (x − 2)² for all real x. Which statement is correct?

• f is decreasing for x < 2
• f has a local minimum at x = 2
• f is decreasing for x > 2
• f is never decreasing

Answer: f is never decreasing

(x − 2)² ≥ 0 everywhere, so f' is never negative and f never decreases. At x = 2 the derivative is zero but does not change sign, so there is no local extremum.

The graph of y = f'(x) is a straight line with positive gradient that crosses the x-axis at x = 2. Which statement about f is correct?

• f has a local minimum at x = 2
• f has a local maximum at x = 2
• f is increasing for all x
• f is decreasing for x > 2

Answer: f has a local minimum at x = 2

f' changes from negative (x < 2) to positive (x > 2), so f changes from decreasing to increasing: a local minimum at x = 2.

Let f(x) = 2x³ + 3x² − 12x. Determine where f is increasing.

• −2 < x < 1
• x > 1 only
• x < −2 or x > 1
• x < −2 only

Answer: x < −2 or x > 1

f'(x) = 6x² + 6x − 12 = 6(x + 2)(x − 1) > 0 when x < −2 or x > 1. Choosing −2 < x < 1 is the classic sign-diagram mix-up: there f' < 0.

A function f has a local maximum at x = c. What does the graph of y = f'(x) do at x = c?

• It has a local maximum
• It crosses the x-axis from negative to positive
• It crosses the x-axis from positive to negative
• It takes the value f(c)

Answer: It crosses the x-axis from positive to negative

At a local maximum f changes from increasing to decreasing, so f' changes sign from positive to negative, crossing the x-axis downwards at x = c.

Let f(x) = x⁴ − 8x². On which of the following intervals is f decreasing?

• −2 < x < 0
• x > 2
• all x < 0
• 0 < x < 2

Answer: 0 < x < 2

f'(x) = 4x³ − 16x = 4x(x − 2)(x + 2). Testing signs: f' < 0 for x < −2 and for 0 < x < 2, and f' > 0 for −2 < x < 0 and x > 2. Of the options, only 0 < x < 2 is a decreasing interval.