Ex 3: If f′(x) = 0 for all x ∈ (a, b), then f(x) = c for some constant c.

Ex 4: If f′(x) = g′(x) for all x ∈ (a, b), then f(x) = g(x) + c for some constant c.

Increasing/Decreasing Test: If f′(x) > 0 for all x ∈ (a, b), then f is increasing on (a, b)

If f′(x) < 0 for all x ∈ (a, b), then f is decreasing on (a, b)

First derivative test: Suppose c is a critical number of a continuous function f, then

Defn: f is concave down if the graph of f lies below the tangent lines to f.

Defn: f is concave up if the graph of f lies above the tangent lines to f.

Concavity Test: If f′′(x) > 0 for all x ∈ (a, b), then f is concave upward on (a, b).

If f′′(x) < 0 for all x ∈ (a, b), then f is concave down on (a, b).

Defn: The point (x_{0}, y_{0}) is an inflection point if f is continuous at x_{0} and if the concavity changes at x_{0}

Second derivative test: If f′′ continuous at c, then If f′(c) = 0 and f′′(c) > 0, then f has a local minimum at c.

If f′(c) = 0 and f′′(c) < 0, then f has a local maximum at c.

If f′(c) = 0 and f′′(c) = 0, second derivative test gives no info.

Converses are not true: Increasing/Decreasing Test If f′(x) > 0 for all x ∈ (a, b), then f is increasing on (a, b) f increasing on (a, b) does not imply f′(x) > 0 for all x ∈ (a, b). Ex:

If f′(x) < 0 for all x ∈ (a, b), then f is decreasing on (a, b) f decreasing on (a, b) does not imply f′(x) < 0 for all x ∈ (a, b). Ex:

Concavity Test: If f′′(x) > 0 for all x ∈ (a, b), then f is concave upward on (a, b). f concave upward on (a, b) does not imply f′′(x) > 0 for all x ∈ (a, b). Ex:

If f′′(x) < 0 for all x ∈ (a, b), then f is concave down on (a, b). f concave downward on (a, b) does not imply f′′(x) < 0 for all x ∈ (a, b). Ex: