4.3 Connecting f’ and f’’ with the graph of f

First derivative test for local extrema f may have local extrema at some critical points while failing to have local extrema at others For a local max, the derivative on the left is positive and the derivative on the right is negative For local min, the derivative on the left is negative and the derivative on the right is positive

Concavity When the derivative is increasing, the function is concave up (y”>0) When the derivative is decreasing, the derivative is concave down (y”<0)

Points of inflection A point where the graph of a function has a tangent line and where the concavity changes y’’=0 and is positive on one side and negative on the other y’ has a local maximum or minimum at a point of inflection

Second derivative test for local extrema The test fails if f’’=0 or if f’’ fails to exist at x = c