Calculus Section 4.3 Determine the concavity of a function

1. Calculus Section 4.3Determine the concavity of a function Intuitively , a graph is concave up if it “holds water”. It is concave down if it will not “hold water”. A graph is concave up when the tangent line is below the graph. It is concave down when the tangent line is above the graph.

2. Test for concavity The graph of f(x) is: Concave down when f”(x) < 0 Concave up when the f”(x) > 0 Where is the function concave up?; concave down? f(x) = x3 – 3x2 + 1

3. A point at which the graph of a continuous function changes concavity is called a point of inflection. The 2nd derivative of a continuous function is equal to zero at a point of inflection.

4. example Find the point(s) of inflection of the function f(x) = x4 + 2x3 – 12x2 + 60x - 8

5. The 2nd Derivative Test can be used to determine whether critical points are relative maximums or minimums. Note: A function is concave down at its local max and concave up at its local min. The Second Derivative Test If (c,f(c)) is a critical point of f(x); 1. If f”(c) < 0, then (c,f(c)) is a relative maximum. 2. If f”(c) > 0, then (c,f(c)) is a relative minimum

6. a. Determine the relative extrema of f(x) = x3 – 6x2 + 9x + 4 using the 2nd derivative test. b. Find the point of inflection. • Graph the function.

7. Sketch the graph of f(x) = 6x2/3 – 4x

8. Federal government expenditures True or false 1. The expenditures at t1 are less than at t2. • The rate at which expenditures are changing at t1 is less than at t2. • The rate of change of expenditures starts decreasing at t2. • The rate of change of expenditures is lower at t3 than at t2. • The expenditures are always increasing.

9. assignment • Page 216 • Problems 2 – 52 even