Derivatives and the Shape of a Graph

Increase, decrease, and local extrema

Concavity and the second derivative

The full curve-sketching protocol

A worked sketch

The derivative f prime is positive on (-3, 0) and negative on (0, 2). What does f do on those intervals?

On an interval, f double prime of x is negative everywhere. What does that say about the graph of f there?

Which condition identifies an inflection point of f at an interior point c?

For f(x) = x^3 - 3x^2 - 9x + 5 with f prime of x = 3(x+1)(x-3), which point is the local maximum?

For the same cubic, f double prime of x = 6x - 6. Where is the graph concave up?

The function f(x) = x^4 has f double prime of 0 = 0, yet the origin is NOT an inflection point. Why not?

For f(x) = x^3 - 3x^2 - 9x + 5, the local maximum occurs at x = -1. What is the value f(-1)? Answer with a number.

The graph of f has f double prime of x = 6x - 6. At what x does the inflection point occur? Answer with a number.

The real line is partitioned by two critical points, x = -1 and x = 3. How many intervals of constant sign for f prime result? Answer with a number.

The cubic f falls on the interval (-1, 3). What is the length of this decreasing stretch? Answer with a number.