Question

determine where $f(x)$ is increasing. $$ f(x)=x^{3} $$

   determine where $f(x)$ is increasing.

$$
f(x)=x^{3}
$$

Calculus for AP

Calculus for AP

Jon Rogawski & Ray… 2nd Edition

Chapter 1, Problem 50

Instant Answer

Solved by Expert Aymara Gallardo

08/06/2019

Step 1

The derivative of a function tells us the rate at which the function is changing at any given point. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing. The derivative of $f(x) = x^{3}$ is $f'(x) = Show more…

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determine where $f(x)$ is increasing. $$ f(x)=x^{3} $$

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Key Concepts

Monotonic Functions

A function is said to be increasing (or monotonic increasing) if, whenever x1 < x2, the function values satisfy f(x1) ? f(x2). This concept is fundamental in understanding the behavior of functions and is used to analyze how the output changes with respect to the input over certain intervals.

First Derivative Test

The first derivative test involves taking the derivative of a function and examining its sign. If the derivative is positive on an interval, it implies that the function is increasing on that interval. Conversely, a negative derivative would indicate that the function is decreasing.

Critical Points

Critical points are where the derivative of a function is zero or does not exist. These points can signal transitions in the behavior of a function, such as changes from increasing to decreasing, and are essential in analyzing the overall monotonicity of a function.

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