In the past, one of the important uses of derivatives was as an aid in curve sketching. We usually use a calculator of computer to draw complicated graphs, it is still important to understand the relationships between derivatives and graphs. 4.3 Connecting f’ and f’’ with the Graph of f

First Derivative Test for Local Extrema at a critical point c 4.3 Connecting f’ and f’’ with the Graph of f 1.If f ‘ changes sign from positive to negative at c, then f has a local maximum at c. local max f’>0f’<0 2. If f ‘ changes sign from negative to positive at c, then f has a local minimum at c. 3.If f ‘ changes does not change sign at c, then f has no local extrema. local min f’<0 f’>0 no extreme f’>0

First derivative: is positive Curve is rising. is negative Curve is falling. is zero Possible local maximum or minimum. 4.3 Connecting f’ and f’’ with the Graph of f

Definition Concavity The graph of a differentiable function y = f(x) is a.concave up on an open interval I if y’ is increasing on I. (y’’>0) b.concave down on an open interval I if y’ is decreasing on I. (y’’<0) concave down concave up

4.3 Connecting f’ and f’’ with the Graph of f Second Derivative Test for Local Extrema at a critical point c 1.If f’(c) = 0 and f’’(c) < 0, then f has a local maximum at x = c. 2.If f’(c) = 0 and f’’(c) > 0, then f has a local minimum at x = c. +

Second derivative: is positive Curve is concave up. is negative Curve is concave down. is zero Possible inflection point (where concavity changes). 4.3 Connecting f’ and f’’ with the Graph of f

Definition Point of Inflection A point where the graph of a function has a tangent line and where the concavity changes is called a point of inflection. inflection point

Set First derivative test: negative positive Possible extreme at. 4.3 Connecting f’ and f’’ with the Graph of f Sketch the graph zeros at x = -1, x = 2

Set First derivative test: maximum at minimum at Possible extreme at. 4.3 Connecting f’ and f’’ with the Graph of f

Set Possible extreme at. Or you could use the second derivative test: maximum atminimum at negative concave down local maximum positive concave up local minimum 4.3 Connecting f’ and f’’ with the Graph of f

We then look for inflection points by setting the second derivative equal to zero. Possible inflection point at. negative positive inflection point at 4.3 Connecting f’ and f’’ with the Graph of f

Make a summary table: rising, concave down local max falling, inflection point local min rising, concave up 4.3 Connecting f’ and f’’ with the Graph of f