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Chapter 2: Problem 31

Sketch a graph of a function whose derivative is always negative.

Short Answer

Expert verified
The graph of a function whose derivative is always negative would be a visually descending line over the point it is defined.

Step by step solution

01

Understand the Concept of Decreasing Functions

In calculus, a function \(f(x)\) is called decreasing on an interval if for any \(x_1\) and \(x_2\) inside the interval, \(f(x_1) \geq f(x_2)\) whenever \(x_1 < x_2\). In other words, as \(x\) increases in value within the interval, \(f(x)\) decreases. This means that the function descends from left to right.
02

Sketch the Graph

Given the understanding of the properties of a decreasing function, it can be easily sketched. Start from a point in the upper part of the graph and draw a line descending to the right. This effectively means that the derivative of this function is always negative, and the function is continually decreasing.
03

Verification

The graph of a decreasing continuous function is always going downhill as we move from left to right. If this condition is met in your sketched graph, then you have correctly drawn a function whose derivative is always negative.

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