43-46= The graph of a polynomial function is given. From the...

$43-46=$ The graph of a polynomial function is given. From the graph, find
(a) the $x$ - and $y$ -intercepts
(b) the coordinates of all local extrema
(GRAPH CAN'T COPY)
$$P(x)=\frac{2}{9} x^{3}-x^{2}$$

Key Concepts

Derivatives and Critical Points

Derivatives provide the rate of change of a function and are used to find critical points, where the derivative is zero or undefined. Critical points are potential candidates for local extrema. By applying the derivative and setting it equal to zero, one can determine where the function may have peaks or valleys, which is a crucial step in understanding the overall behavior of the graph.

Local Extrema

Local extrema, which include local maxima and local minima, are points on the graph where the function reaches a highest or lowest value within a given small neighborhood. These points are significant in analyzing the behavior of the function as they indicate changes in the direction of the graph, and are identified by evaluating the function and its derivative.

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of the variables. These functions are continuous and smooth, which makes their graphs predictable in behavior, especially when analyzing intercepts and turning points.

Intercepts

Intercepts are points where the graph of a function crosses the coordinate axes. The x-intercepts are found by setting the function equal to zero and solving for the variable, while the y-intercept is determined by evaluating the function at zero. Understanding intercepts is fundamental when sketching a function since they provide key points that the graph must pass through.

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