Find the interval(s) where the function is increasing and the interval(s) where it is decreasing

step 1 The function is fx equals to 2 divide by 3 multiplied by x cube minus 2x squared minus 6x minus

Find the interval(s) where the function is increasing and...

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Key Concepts

Derivative

The derivative of a function gives the slope of the tangent line at any point and is a fundamental tool for analyzing the behavior of a function. It helps determine whether a function is increasing or decreasing by indicating the rate of change of the function with respect to its independent variable.

Critical Points

Critical points are the values of the independent variable where the derivative of the function is zero or undefined. These points serve as potential boundaries between intervals of increasing and decreasing behavior, and identifying them is essential for subsequent analysis using the derivative.

First Derivative Test

The first derivative test involves analyzing the sign of the derivative on intervals defined by the critical points to decide whether the function is increasing (derivative positive) or decreasing (derivative negative). This method is a key technique in determining the monotonicity of a function.

Increasing and Decreasing Intervals

These intervals represent the segments of the domain where the function is consistently rising (increasing) or falling (decreasing). Determining these intervals through the sign analysis of the derivative is a central concept in understanding the overall behavior and trends of a function.

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