/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 The graph of \(f^{\prime}\) is g... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 38

The graph of \(f^{\prime}\) is given. Draw a rough sketch of the graph of \(f\) given that \(f(0)=1.\)

Short Answer

Expert verified
The sketch of the function \(f\) should show an increasing or decreasing function depending on the derivative function \(f'\). The sketch should also show local maxima or minima wherever \(f'\) crosses zero. The entire sketch is anchored at the point where \(x=0\), \(f(0)=1\). Note that without the actual graph of \(f'\), it's not possible to provide a more precise description of the resulting graph of \(f\).

Step by step solution

01

Identify the regions where \(f\) is increasing and decreasing

Observe the graph of \(f'\). The function \(f\) will be increasing wherever \(f'\) is positive and decreasing wherever \(f'\) is negative.
02

Identify the local maxima or minima

Find the points where \(f'\) crosses zero. These are the points where the function \(f\) has local maxima or minima. Note whether the derivative changes sign from positive to negative (local maxima) or negative to positive (local minima).
03

Identify the function value at \(x=0\)

The value of the function \(f\) at \(x=0\) is given as \(f(0)=1\). This serves as an anchor point for sketching the function.
04

Sketch the graph of the function

Using the information about the intervals of increase and decrease, the local maxima or minima and the value at \(x=0\), draw a rough sketch of the function \(f\). Carefully make sure to observe the qualitative aspects of the graph: when \(f\) is increasing/decreasing, where the local maxima or minima are located and how the function behaves around the point \(x=0\)

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