What Derivatives Tell us

The function f (x) = 3x² is

• (a) increasing on (−∞, 0) and decreasing on (0, ∞).
• (b) increasing on (−∞, ∞).
• (c) decreasing on (−∞, 0) and increasing on (0, ∞).

A function f has the property that f ′(x) > 0 on (a, b) and f ′(x) < 0 on (b, c). It follows that

• (a) f is continuous on (a, c).
• (b) f is increasing on (a, c).
• (c) f is decreasing on (b, c).

A function f has the property that f ′(x) > 0 on (a, b), f ′(x) < 0 on (b, c), and f ′(b) = 0 . It follows that

• (a) f has a local maximum at b.
• (b) f has a local minimum at b.
• (c) f is undefined at b.

If f ′ is decreasing on an interval I, then

• (a) f is decreasing on I.
• (b) f is concave down on I.
• (c) f is concave up on I.

If f ′′(c) = 0 , then

• (a) an inflection point occurs at x = c.
• (b) the concavity of f changes at c.
• (c) c is a candidate for the location of an inflection point.

If f is concave down on an interval I, then

• (a) f is decreasing on I.
• (b) f ′′(c) < 0 on I.
• (c) f is positive on I.

The function f has the property that f ′(c) = 0 and f ′′(c) > 0 . It follows that

• (a) f has a local minimum at x = c.
• (b) f has a local maximum at x = c.
• (c) f is increasing on an interval containing c.

The graph of a function f that satisfies f ′(x) > 0 and f ′′(x) < 0 on (0, ∞)

• (a) increases at an increasing rate.
• (b) increases at a decreasing rate.
• (c) decreases at a decreasing rate.

Which is true of f (x) = sin x on (0, π)?

• (a) It is increasing and concave down.
• (b) It is decreasing and concave down.
• (c) It is positive and concave down.

The function f (x) = x⁸ satisfies f (0) = f ′(0) = f ′′(0) = 0 . Which is true?

• (a) A local minimum and an inflection point occur at x = 0.
• (b) A local minimum occurs at x = 0.
• (c) An inflection point occurs at x = 0.