Graph the polynomial in the given viewing rectangle. Find...

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

Key Concepts

Polynomial Functions

A polynomial function is an expression involving variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponents. They are continuous, smooth functions that exhibit typical end behavior based on their leading terms, which is crucial when predicting the overall behavior of the graph.

Graphing Functions in a Specified Viewing Rectangle

This concept involves plotting a function within a defined region of the coordinate plane. The specified viewing rectangle restricts the x and y values visible on the graph, aiding in accurately depicting key features such as intercepts, turning points, and behavior near the boundaries, which is essential for a clear visualization.

Derivatives and Critical Points

The derivative of a function represents its rate of change and is a fundamental tool in calculus for analyzing the behavior of functions. Critical points are found by setting the derivative equal to zero or identifying where the derivative does not exist. These points are where the function may have a local maximum or minimum, which helps in determining the local extrema.

Local Extrema

Local extrema refer to points on the function where a local maximum (highest value) or local minimum (lowest value) occurs in a neighborhood around those points. Finding these involves computing the derivative to locate critical points and then applying tests such as the first or second derivative test to classify them as local maxima or minima.

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