These exercises involve local maxima and minima of...

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.

Key Concepts

Graphing Polynomial Functions

Graphing polynomial functions involves using the roots (or zeros) found by factorization, together with information about the degree and leading coefficient, to determine the overall shape of the graph. For polynomials expressed as a product of linear factors, the zeros indicate where the graph intersects the x-axis, while the changes in direction between these zeros indicate the presence of local extrema.

Polynomial Functions

These are functions defined by a polynomial expression, which is a sum of powers of x with constant coefficients. They are continuous and differentiable everywhere, making them central subjects in calculus for studying behavior such as turning points and end behavior.

Cubic Polynomials

Cubic polynomials are third-degree polynomials that typically exhibit an ‘S’ shaped curve. Due to their degree, they can have up to two turning points, which correspond to local maximum and minimum values, and their general behavior includes a change in concavity.

Local Extrema

Local extrema refer to the local high (maximum) or low (minimum) points on a function’s graph where the function value is greater than or less than the values of the function at nearby points. For continuous differentiable functions, these are identified by points where the function’s derivative equals zero and exhibits a change in sign.

Critical Points

Critical points of a function occur where its derivative is zero or undefined. For differentiable functions like polynomials, these points are key in identifying potential local extrema, and the nature of these points can often be confirmed with derivative tests or by analyzing the graph.

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