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.
$$
y=x^{3}-12 x+9, \quad[-5,5] \text { by }[-30,30]
$$

Key Concepts

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients combined using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. These functions are smooth, continuous, and differentiable everywhere, and their degree determines their end behavior and the maximum number of turning points they may have.

Derivative

The derivative of a function represents its instantaneous rate of change and is a fundamental tool in calculus for analyzing the behavior of functions. It is used to determine slopes at any given point, which is essential for identifying critical points and understanding increasing or decreasing intervals of the function.

Critical Points

Critical points are values in the domain of a function where the derivative is zero or undefined. These points are important because they are potential locations for local extrema, where the function might change from increasing to decreasing or vice versa.

Local Extrema

Local extrema refer to the local maximum and minimum values of a function, which occur at critical points. A local maximum is a point where the function value is higher than those of nearby points, while a local minimum is lower than those of adjacent points. Identifying these points is crucial in understanding the overall behavior of the function.

Graphing in a Viewing Rectangle

Graphing within a specified viewing rectangle involves plotting the function over a limited domain and range to capture relevant features of the graph. This practice helps in visualizing the behavior of the function, including its intercepts, turning points, and overall shape, within a predetermined window of interest.

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