Question

determine where $f(x)$ is increasing. $$ f(x)=x^{4} $$

   determine where $f(x)$ is increasing.

$$
f(x)=x^{4}
$$

Calculus for AP

Calculus for AP

Jon Rogawski & Ray… 2nd Edition

Chapter 1, Problem 51

Instant Answer

Solved by Expert Linh Vu

03/03/2020

Step 1

The derivative of a function gives us the slope of the tangent line at any point, which can tell us whether the function is increasing or decreasing. The derivative of $f(x) = x^{4}$ is $f'(x) = 4x^{3}$. Show more…

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determine where $f(x)$ is increasing. $$ f(x)=x^{4} $$

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Key Concepts

Derivative

The derivative of a function represents its instantaneous rate of change with respect to the input variable. In calculus, taking the derivative is a primary tool used to analyze where the function is increasing or decreasing. For a function to be increasing, its derivative must be positive. This concept is central to understanding the behavior of differentiable functions.

Monotonicity

Monotonicity refers to the property of a function to be consistently increasing or decreasing over an interval. A function is considered increasing if, for any two numbers within that interval where the first is less than the second, the function's value at the first is less than or equal to its value at the second. The analysis of monotonicity is fundamental in calculus, especially when determining the behavior of functions over their domains.

Critical Points

Critical points are points in the domain of a function where the derivative is either zero or undefined. These points are important because they indicate potential changes in the function's monotonicity. By identifying critical points, one can divide the domain into distinct intervals and analyze the behavior of the function -- specifically, whether it is increasing or decreasing in each interval.

Sign Analysis of the Derivative

After computing the derivative of a function, the next step in analyzing its increasing or decreasing behavior is performing a sign analysis. This involves determining the intervals where the derivative is positive, which corresponds to intervals where the function is increasing, and where it is negative, which corresponds to intervals of decrease. This method is an essential part of applying the first derivative test in calculus.

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