Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Determine the extrema on the given interval.$$f(x)=x /\left(x^{2}+1\right) \text { on: }(a)[0,2] ;(b)[-2,2] ;(c)(0,2) ;(d)(-2,2)$$

(a) $\mathrm{M}=^{1} / 2, \mathrm{m}=0$(b) $\mathrm{M}=^{1} / 2, \mathrm{m}=-1 / 2$(c) $\mathrm{M}=^{1} / 2,$ no min(d) $\mathrm{M}=^{1} / 2, \mathrm{m}=-1 / 2$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 1

Extrema of a Function

Derivatives

Harvey Mudd College

Baylor University

University of Nottingham

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

15:39

Find the absolute extrema …

01:00

Locate the absolute extrem…

02:35

01:24

01:55

Determine the extrema on t…

02:19

Numerically estimate the a…

02:25

01:50

Finding Extrema on an Inte…

03:23

02:40

02:00

03:14

Yeah. First we want to find the critical points of this function. We take a derivatives to this function. The denominator is just the square of the pro real strong meter. The numerator equals derivative of the numerator times of the denominator miners. The numerator times the derivative of the dominator. And the numerator becomes minors access greer minus one. And we can factor the numerator into two parts X plus Y and X miners. You feel that that directive equals zero. Then we get to critical points. Minor swan. Yeah. No one apart A. We want to find an extreme of this functioning between the interval zero and two. The critical point why is raising this syndrome enhanced? They varied their function values, Rachel their role half to fifth and there therefore there was the minimum and a half is a maximum. Yeah. And for part B. We're going to find the extreme of this function between the interval minus two energy. Now both minors one and one have visited this interest and they varied their functioning numbers hr minus trophies. My nurse to have half and to offense. Therefore miners half the minimum and a half is an accident. Now for piracy the interval is the same but it's an open interest so it cannot reach the value zero oh to fifth. So it has no minimum. But it still has a maximum. It was a heart. Have a party. It's the salmon per B body. It is in or paint. Hence it cannot reach the value miners to fifth or to fifth but it can still richer value or half are minors. Therefore miners are half is to its minimum and behalf as its maximum.

View More Answers From This Book

Find Another Textbook

01:15

Suppose that a function satisfies the same conditions as in the previous exe…

01:09

Sketch the graph of a function which is decreasing to the left of $x=1,$ inc…

02:59

Use the first derivative to determine where the given function is increasing…

03:09

Find $\frac{d}{d x}(\sqrt{2 x+\sqrt{3 x+\sqrt{4 x}}})$.

02:44

If a manufacturer has fixed costs of $700,$ a cost per item for production o…

Use the method developed in the previous exercise to find the equation of th…

03:24

An alternative way to do implicit differentiation is to introduce a mythical…

02:18

Find the average rate of change of $y$ with respect to $x$ on the given inte…

01:04

Given $H(x)=\left\{5\left(x^{2}-4\right)^{3}+\left(x^{2}-4\right)^{10}\right…

03:35

At a selling price of $50,$ a lamp company breaks even on total sales of $30…