Local Extrema These exercises involve local maxima and...

Local Extrema These exercises involve local maxima and minima of polynomial functions.
(a) Graph the function $P(x)=(x-2)(x-4)(x-5)$ and
determine how many local extrema it has.
(b) If $a<b<c$, explain why the function
$$
P(x)=(x-a)(x-b)(x-c)
$$
must have two local extrema.

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Key Concepts

Local Extrema

Local extrema are points in a function's domain where the function reaches a local maximum or minimum value. These points are important because they indicate where the function changes its direction of growth from increasing to decreasing or vice versa. They are typically found by analyzing points where the rate of change (derivative) of the function is zero or undefined.

Critical Points

Critical points are points in the domain of a function where its derivative is either zero or does not exist. In the context of smooth polynomial functions, these points are where potential local extrema occur. Determining the critical points is a key step in analyzing the behavior of the function and understanding its graph.

Polynomial Function Behavior

Polynomial functions are smooth, continuous, and differentiable functions characterized by their degree. The degree of the polynomial dictates the maximum number of turning points, which is at most one less than the degree. This inherent behavior helps in predicting the number of local extrema that the function can have.

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