Local Extrema Graph the polynomial in the given viewing...

Local Extrema Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. State the domain and range.
$$
y=x^{4}+4 x^{3},[-5,5] \text { by }[-30,30]
$$

Key Concepts

Graphing within a Viewing Rectangle

Graphing within a viewing rectangle means plotting the function only over a specified interval for x and y values. This practice focuses on a region of interest, allowing one to analyze key features such as local extrema within that bounded domain and range while ignoring behavior outside of it.

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of powers of x with constant coefficients. They form smooth, continuous curves on a graph and their general behavior, such as end behavior and turning points, is determined by the degree and the coefficients of the polynomial.

Local Extrema

Local extrema refer to the points on a function where its value is either locally highest (local maximum) or locally lowest (local minimum) in a neighborhood around the point. They are critical for understanding the function's behavior and are typically found by setting the derivative equal to zero and analyzing the resulting critical points.

Domain and Range

The domain of a function is the set of all input values (x-values) for which the function is defined, while the range is the set of all output values (y-values) that the function can produce. In the context of a restricted viewing rectangle, the domain and range correspond to the x- and y-intervals given, highlighting only the portion of the function being examined.

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