# Find Local Maxima/Minima of y=f'(x) from y=f(x)

• undrcvrbro
In summary, the process for locating local maxima and minima for the graph of y=f'(x) by examining the graph of y=f(x) involves identifying inflection points on the graph of y=f(x) and observing where the concavity changes. This change in concavity indicates a change from increasing to decreasing or vice versa for f'(x), making the point of change a local extremum for f'(x). This method can be used instead of finding the second derivative to locate the inflection points.
undrcvrbro

## Homework Statement

Explain how you can locate the local maxima and minima for the graph of y=f '(x) by examining the graph of y=f(x).

## The Attempt at a Solution

If there is an inflection point on the graph of y=f(x) at x=c, then f(x) must change concavity at x=c. Consequently, f '(x) must change from increasing to decreasing or from decreasing to increasing at x=c, and x=c is a local extremum for f '(x). If there is an inflection point on the graph of y=f(x) at x=c, then f(x) must change concavity at x=c. Consequently, f '(x) must change from increasing to decreasing or from decreasing to increasing at x=c, and x=c is a local extremum for f '(x).

I must be missing something. Don't you need to know the second derivative in order to know where the inflection points actually are?

Maybe they're just asking you to locate, by actually plotting y=f(x), its inflection points.

Well, not by plotting a specific function, but just explaining that at an inflection point, the graph changes from "convex up" to "convex down". Typically we find the second derivative, in order to find the inflection points, in order to tell where the curve changes convexity. The point of this problem is we can do it, at least roughly, the other way. If we look at the graph and can see where it changes convexity, we can see where the inflection points are (and, so, where the second derivative is 0).

oh

ohh alright. thanks ivy.

## What is the purpose of finding local maxima/minima of y=f'(x) from y=f(x)?

The purpose of finding local maxima/minima of y=f'(x) from y=f(x) is to identify the points on a graph where the slope (derivative) of a function is equal to zero. These points can provide important information about the behavior of the function, such as the location of peaks and valleys, and can be used to optimize functions in various real-world applications.

## How do you find the local maxima/minima of y=f'(x) from y=f(x)?

The local maxima/minima of y=f'(x) from y=f(x) can be found by taking the derivative of the original function, setting it equal to zero, and solving for the value(s) of x. These values represent the critical points of the function, and can be further analyzed to determine if they are local maxima or minima.

## What is the difference between local and global maxima/minima?

The main difference between local and global maxima/minima is that local maxima/minima are points on a graph where the slope (derivative) of a function is zero within a certain interval, while global maxima/minima are points where the function has the highest or lowest value over the entire domain. In other words, local maxima/minima are relative extremes, while global maxima/minima are absolute extremes.

## Can a function have more than one local maxima/minima?

Yes, a function can have multiple local maxima/minima. This occurs when the derivative of the function is equal to zero at multiple points within a given interval. In this case, each of these points represents a local maxima or minima, and further analysis is needed to determine which of these points is the global maxima or minima.

## How are local maxima/minima used in real-world applications?

Local maxima/minima are used in a variety of real-world applications, such as optimizing production processes, maximizing profits, and minimizing costs. By identifying the points where a function's derivative is equal to zero, these applications can make informed decisions to improve efficiency and achieve their desired outcomes.

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