Point of Inflection and Extrema Show that the point of inflection of f(x)=x(x-6)

Step 1: Find the first derivative of the function ( f(x) = x(x-6)^2 ). Using the product rul

Point of Inflection and Extrema Show that the point ...

$\begin{array}{l}{\text { Point of Inflection and Extrema Show that the point }} \\ {\text { of inflection of }} \\ {f(x)=x(x-6)^{2}} \\ {\text { lies midway between the relative extrema of } f .}\end{array}$

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Key Concepts

Inflection Point

An inflection point is a point on the graph of a function at which the concavity changes from upward (concave up) to downward (concave down), or vice versa. This change is typically identified by examining when the second derivative is zero and confirming that a sign change occurs, indicating a transition in the function's curvature.

Critical Points and Relative Extrema

Critical points occur where the first derivative of a function is zero or undefined. These points are examined to determine relative extrema, which are local maximums and minimums. Using tests like the first derivative test or the second derivative test, one can ascertain whether these critical points correspond to peaks or valleys in the function's graph.

Derivative

A derivative measures the instantaneous rate of change of a function, which is essential for analyzing its behavior. It helps in identifying intervals where the function is increasing or decreasing and is used to locate critical points by setting the derivative equal to zero. These critical points often serve as candidates for local maxima or minima.

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