Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.
7. y=sin|x|, -2π≤x≤2π

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To find the inflection points, local maxima, and minima, first consider the function y = sin|x|. The absolute value affects the symmetry of the graph, making it even.

Identify critical points by finding where the derivative is zero or undefined. The derivative of y = sin|x| is y' = cos|x| * (x/|x|), which is undefined at x = 0.

Evaluate the second derivative to determine concavity. The second derivative will help identify intervals where the graph is concave up or concave down.

Analyze the graph visually to identify local maxima and minima. The peaks and troughs of the sine wave indicate these points, typically at x = ±π, ±2π.

Determine the open intervals of concavity. The graph is concave up where the second derivative is positive and concave down where it is negative. This typically alternates between intervals of π.

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Inflection Points

Inflection points occur where the concavity of a function changes from concave up to concave down or vice versa. To find these points, examine where the second derivative changes sign. For the function y = sin|x|, analyze the graph and derivatives to identify these transitions within the given interval.

Local Maxima and Minima

Local maxima and minima are points where a function reaches a peak or a trough, respectively, within a certain interval. These can be found by setting the first derivative to zero and analyzing the sign changes. For y = sin|x|, observe the graph to determine where the function reaches its highest and lowest values within the specified range.

Concavity and Differentiability

Concavity describes the direction a graph curves, either upwards (concave up) or downwards (concave down). Differentiability refers to the ability to compute a derivative at a point. For y = sin|x|, identify intervals of concavity by examining the second derivative and check differentiability by ensuring the function is smooth and continuous over the given domain.

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.7. y=sin|x|, -2π≤x≤2π