   Chapter 3.3, Problem 49E

Chapter
Section
Textbook Problem

# 49-50(a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection.(b) Use a graph of f′′ to give better estimates. f ( x ) = sin 2 x + sin 4 x ,   0 ≤ x ≤ π

To determine

(a)

To find:

To give rough estimates of f by using the graph on:

(i) The intervals of concavity

(ii) Coordinates of the point of inflection

Explanation

1) Concept:

Sketch the graph of f and find the intervals of concavity and coordinates of the point of inflection using its definition.

2) Definition:

Inflection point: A point P on curve y=f(x) is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward

3) Given:

fx=sin 2x+sin 4x,   0xπ

4) Calculation:

Consider the function,

fx=sin 2x+sin 4x,   0xπ

The graph of f is,

By observing the graph,

The curve is concave upward on intervals: (0.8, 1.6) and (2.3, π)

The curve is concave downward on intervals: (0, 0.8) and (1

To determine

(b)

To find:

By using the graph of f" the better estimates of,

(i) The intervals of concavity

(ii) coordinates of the point of inflection

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