Find all critical values, the largest open intervals on which f is increasing, the largest open

step 1 we have a function of x that is absolute value x plus 3 minus 1 and before that we find the crit

Find all critical values, the largest open intervals on...

Key Concepts

Graphing Polynomial Functions

Sketching the graph of a polynomial function involves analyzing its critical points, increasing/decreasing intervals, and relative extrema, along with understanding its overall end behavior. For cubic functions, this analysis provides insight into the smooth, continuous nature of the curve and helps guide the construction of a rough sketch that reflects the function's behavior across its domain.

Critical Points

Critical points are values in the domain of a function where its derivative is zero or undefined. These points are crucial because they can indicate where the function may have local maximum or minimum values, or where the function’s behavior changes. In the context of calculus, finding the critical points is the first step in analyzing the overall behavior of a function.

First Derivative

The first derivative of a function represents the instantaneous rate of change or the slope of the tangent at a point. Calculating the first derivative and setting it equal to zero is essential for finding the critical points of a function. Analyzing the sign of the first derivative on different intervals helps determine where the function is increasing or decreasing.

Increasing and Decreasing Intervals

These intervals are determined by the sign of the first derivative. When the derivative is positive over an interval, the function is increasing; when it is negative, the function is decreasing. Identifying these intervals is important for understanding the overall behavior of the function and for locating relative extrema.

Relative Extrema

Relative extrema refer to the local maximum and minimum values of a function. They occur at critical points where there is a change from increasing to decreasing (yielding a local maximum) or decreasing to increasing (yielding a local minimum). Recognizing these points is key in optimization problems and in understanding the function’s graph.

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