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.
$$
y=x^{5}-5 x^{2}+6, \quad[-3,3] \text { by }[-5,10]
$$

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving sums of powers of x with constant coefficients. They are continuous, differentiable functions that are used to model a variety of behaviors in algebra and calculus, and their properties such as end behavior and turning points can be analyzed using calculus tools.

Local Extrema

Local extrema refer to the points on a graph where the function reaches a local maximum or minimum value. These points occur where the function changes direction, and are typically found by identifying where the derivative is zero or undefined, then applying tests to confirm the nature of these points.

Critical Points

Critical points are values of x in the domain of a function where its derivative is zero or does not exist. These points are essential in determining where local extrema might occur, as a change in the sign of the derivative around these points indicates a transition from increasing to decreasing behavior, or vice versa.

Derivative Test

Derivative tests, including the first derivative test, are techniques used to analyze the change in sign of the derivative before and after a critical point. By determining whether the derivative shifts from positive to negative or negative to positive, one can decide if the point is a local maximum or minimum.

Domain and Range

The domain of a function is the complete set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values). Understanding domain and range is essential for describing the behavior and restrictions of a function, especially when it is analyzed within particular intervals or viewing windows.

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