/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Explain how the Second Derivativ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 6

Explain how the Second Derivative Test is used.

Short Answer

Expert verified
Answer: The six steps to using the Second Derivative Test are: 1) Understand the concepts of local minimums, local maximums, and saddle points. 2) Find the first derivative of the function. 3) Identify the critical points. 4) Find the second derivative of the function. 5) Evaluate the second derivative at the critical points. 6) Use the Second Derivative Test to classify the critical points as local minimums, local maximums, or saddle points.

Step by step solution

01

Understanding the concepts of local minimums, local maximums, and saddle points

To use the Second Derivative Test, we first need to understand the basic terminology. - A local minimum is a point on the graph of a function where the function's value is lower than any other point in its neighborhood. - A local maximum is a point on the graph of a function where the function's value is higher than any other point in its neighborhood. - A saddle point is a point on the graph of a function where neither a local minimum nor a maximum occurs, and the function may change its concavity at that point.
02

Find the first derivative of the function

Given a continuous function f(x), we need to find the first derivative f'(x). To do this, we differentiate f(x) with respect to x. Depending on the function, we may need to apply basic differentiation rules or use more advanced techniques like the chain rule or implicit differentiation.
03

Identify the critical points

The next step is to identify the critical points of the function. Critical points are where the first derivative is equal to zero or undefined. To find the critical points, we solve the equation f'(x) = 0 and/or identify values of x for which f'(x) is undefined.
04

Find the second derivative of the function

To use the Second Derivative Test, we need the second derivative f''(x). We find this by differentiating the first derivative f'(x) with respect to x.
05

Evaluate the second derivative at the critical points

For each critical point found in step 3, we evaluate the second derivative f''(x) at these points.
06

Use the Second Derivative Test to classify the critical points

Based on the value of the second derivative at each critical point, we can classify the point as a local minimum, local maximum, or saddle point: - If f''(x) > 0 at the critical point, then the point is a local minimum because the function is concave up. - If f''(x) < 0 at the critical point, then the point is a local maximum because the function is concave down. - If f''(x) = 0 at the critical point, the Second Derivative Test is inconclusive. In this case, we may need to use other methods, such as analyzing the first derivative or using the First Derivative Test. By following these steps, we've successfully used the Second Derivative Test to classify critical points of a given function.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The temperature of points on an elliptical plate \(x^{2}+y^{2}+x y \leq 1\) is given by \(T(x, y)=25\left(x^{2}+y^{2}\right) .\) Find the hottest and coldest temperatures on the edge of the elliptical plate.

Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension, some types of wave motion are governed by the one-dimensional wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u(x, t)\) is the height or displacement of the wave surface at position \(x\) and time \(t,\) and \(c\) is the constant speed of the wave. Show that the following functions are solutions of the wave equation. \(u(x, t)=A f(x+c t)+B g(x-c t),\) where \(A\) and \(B\) are constants and \(f\) and \(g\) are twice differentiable functions of one variable

In its many guises, the least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe size) in the form of pairs \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) The data may be plotted as a scatterplot in the \(x y\) -plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. Generalize the procedure in Exercise 70 by assuming that \(n\) data points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) are given. Write the function \(E(m, b)\) (summation notation allows for a more compact calculation). Show that the coefficients of the best-fit line are $$ \begin{aligned} m &=\frac{\left(\sum x_{k}\right)\left(\sum y_{k}\right)-n \sum x_{k} y_{k}}{\left(\sum x_{k}\right)^{2}-n \sum x_{k}^{2}} \text { and } \\ b &=\frac{1}{n}\left(\sum y_{k}-m \Sigma x_{k}\right) \end{aligned}, $$ where all sums run from \(k=1\) to \(k=n\).

Find the points (if they exist) at which the following planes and curves intersect. $$y=2 x+1 ; \quad \mathbf{r}(t)=\langle 10 \cos t, 2 \sin t, 1\rangle, \text { for } 0 \leq t \leq 2 \pi$$

A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steadystate distribution of heat in a conducting medium. In two dimensions, Laplace's equation is $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0.$$ Show that the following functions are harmonic; that is, they satisfy Laplace's equation. $$u(x, y)=e^{-x} \sin y$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.