In Exercises 45-52, find the derivative at each critical point and determine the local extreme v

step 1 For this problem, we have been given a piecewise function. Y equals 3 minus X for X values less

In Exercises 45-52, find the derivative at each critical...

Key Concepts

Local Extreme Values

Local extreme values, including local minima and maxima, occur at points where the function takes on the smallest or largest values in a neighborhood. Determining these requires evaluating the function at critical points and possibly at the boundaries of intervals, especially when dealing with piecewise-defined functions.

Critical Points

Critical points are points in the function’s domain where the derivative is zero or fails to exist. These points are vital in identifying potential local maxima and minima, which are important in optimization and curve analysis.

Continuity and Differentiability at Boundaries

When a piecewise function changes its definition at certain points, analyzing continuity and differentiability at those boundary points is essential. This involves checking whether the function value and the derivatives from both sides match, ensuring a smooth transition and identifying any discontinuities or cusps that may influence the existence of local extreme values.

Piecewise-Defined Functions

A piecewise-defined function is one that is specified by different expressions over different intervals of the independent variable. These functions are important in calculus because their behavior at the boundaries (where the defining rules change) can lead to discontinuities or problems with differentiability, making it crucial to analyze each segment separately.

Derivative

The derivative represents the instantaneous rate of change of a function and is a fundamental concept in differential calculus. It is used to understand how the function behaves locally, and its calculation may vary for each piece of a piecewise-defined function, requiring careful attention to the formulas used in different intervals.

Transcript

00:01 For this problem, we have been given a piecewise function.

00:04 Y equals 3 minus x for x values less than 0, and 3 plus 2x minus x squared for x values greater than are equal to 0.

00:13 We're going to find the derivative of this function, find critical points, and see what any local extreme values are.

00:20 Now, as a reminder, critical points occur at two potential different places in a function.

00:27 First, we want to look at anywhere where the derivative of the function, equals 0, or we want to look anywhere where the derivative is undefined.

00:37 Okay? so for our particular function, we're going to find the derivative, and then we're going to examine any points where the derivative is zero or it's undefined.

00:47 Now, just as a reminder, all extreme values are going to happen at critical points, but not every critical point is an extreme value.

00:55 So we'll need to be a little bit discerning.

00:58 Once we find our critical points, we'll have to examine our functional.

01:01 And see what's going on at those points.

01:04 Okay, now, first thing, when we're looking at our function, i know that i'm going to want to examine where they meet.

01:12 I'm just going to kind of put this in parentheses here.

01:15 Because where two pieces join of a function, it's possible that my derivative won't be defined there if the derivative on either side doesn't match up, if it's not a nice smooth curve.

01:26 So i know i'm going to want to look at that point.

01:29 Secondly, let's take a look and make sure that we're ever actually continuous.

01:33 When x equals zero, i use the bottom to find the value of my function, and that gives me a value of three.

01:40 Now, if i look at the top piece when x is zero, if i let x equals zero in that case, it would also give me three.

01:47 So this is a continuous function.

01:50 Does not necessarily mean my derivatives defined there, but it's continuous.

01:53 So i don't have to worry about a gap in my function.

01:58 Let's take the derivative.

02:01 I have it when x is less than 0 when x is greater than 0, and we'll look at when x equals 0 in a minute.

02:08 Okay, first piece, derivative of my first piece of my function is negative 1.

02:13 The derivative of the second piece is 2 minus 2x.

02:18 Okay, so constant derivative for all values of x less than 0, which makes sense.

02:23 That first piece is a line, so constant slope...

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