5. Graphical Applications of Derivatives

For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.

Determine the intervals for which the function is concave up or concave down. State the inflection points. 

$f(x)=2\(\sin\) x+3x$; $0$ < $x$ < $2\(\pi\)$

103. A function f(x) has domain (-2, 2). The graph below is a plot of the derivative of f, not a plot of f itself. In other words, this is a graph of y = f'(x). Either use this graph to determine on which intervals the graph of f is concave up and on which intervals the graph of f is concave down, or explain why this information cannot be determined from the graph.

Each of Exercises 89–92 shows the graphs of the first and second derivatives of a function y=f(x). Copy the picture and add to it a sketch of the approximate graph of f, given that the graph passes through the point P.

131. Let f(x) = x * e^(−x).

b. Find all inflection points for f.

Concavity of parabolas Consider the general parabola described by the function f(x) = ax² + bx + c. For what values of a, b, and c is f concave up? For what values of a, b, and c is f concave down?