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.
$$
y=2 x^{3}-3 x^{2}-12 x-32,[-5,5] \text { by }[-60,30]
$$

Key Concepts

Graphing in a Viewing Rectangle

Graphing a function within a specified viewing rectangle involves plotting the function over a given interval for both the x-axis and the y-axis. This process helps in visualizing important features of the function, such as intercepts, turning points, and the relative positions of local extrema, and ensures that these features are clearly visible in the context of the problem.

Critical Points

Critical points of a function are locations where the derivative is zero or undefined. For smooth functions like polynomials, these points are found by setting the derivative equal to zero. Critical points are important because they are potential locations for local maxima or minima and are foundational in the study of calculus.

Local Extrema

Local extrema refer to the local maximum and minimum values of a function, which are the points where the function reaches its highest or lowest values in a particular neighborhood. Determining local extrema typically involves finding critical points and then using tests, such as the first or second derivative test, to classify these points.

Polynomial Functions

A polynomial function is an expression consisting of variables raised to non-negative integer powers and multiplied by coefficients, combined using addition, subtraction, and multiplication. Understanding polynomial functions is essential because they are continuous, differentiable, and their graphs exhibit smooth curves whose behavior is influenced by the degree and leading coefficient of the polynomial.

Differentiation and Derivatives

Differentiation is the process of finding the derivative of a function, which provides the rate at which the function’s value changes with respect to changes in the input. For polynomial functions, differentiation is particularly straightforward and is used to identify points of interest such as where the function’s slope is zero, which can indicate local extrema.

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