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

(a) Show that $f(x)=\left(x-a_{1}\right)^{2}+\left(x-a_{2}\right)^{2}$ has its minimum value when $x=\left(a_{1}+a_{2}\right) / 2$. (b) Show that $f(x)=\left(x-a_{1}\right)^{2}+\left(x-a_{2}\right)^{2}$ $+\left(x-a_{3}\right)^{2}$ has its minimum value when $x=\left(a_{1}+a_{2}+a_{3}\right) / 3$ (c) Generalize the above.

$$125 \mathrm{ft}$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 4

Applications I - Geometric Optimization Problems

Derivatives

Campbell University

Baylor University

University of Nottingham

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.

01:24

Let $f: R \square R$ be su…

02:25

Let $f(x)=a x^{2}+b x+c,$ …

01:34

Let $f(x)=1+3 x^{2}+3^{2} …

03:43

Consider $f(x)=A x^{3}+B x…

02:19

Show that $f(x)=x^{3}+b x^…

I want us to show that the minimum of dysfunction is just the average of these two values. So we can take the derivative get this so that they could X one uh optimal solutions that are all zero. And we can clearly see that, you know, are to cancel out. We get two x one and then move these to the other side. We get a one plus a two and then that would all be over to. Now. They asked us if we have three, so X minus a one square plus x minus +82 squared plus x minus +83 square. And again we can take the derivative disrespect to X. So X where X one and we get this expression here and r two's cancel out everywhere. And then we wind up with three x one and then a one plus a two plus +83 on the other side. So again we get x one. The optimal solution of this is the average of these numbers here. And I want us to generalize it and that's not hard. You can see the pattern here. So we have f of x equals x minus a one square plus all the way up to x minus eight to the end square. So we have a summation here, um from one to end. And now uh we can take the derivative of that. And each term basically we're going to get two, x minus x one minus a one. And we said x x one are optimal solution plus a bunch of terms that looked like this all the way up to two times x minus AM. Again the two is all cancel out. And then we have if we we see, we have and values of X here X one that we're gonna add up to one there, make it red to get one there. And so we have and values of X that's gonna add up. And then we have all these other A values go to the other side. So we just get a summation of A's. And so in the end we get X one equals again the average of all these constant values.

View More Answers From This Book

Find Another Textbook

Numerade Educator

01:09

Normally if we say a limit $L$ exists, we mean that $L$ is a finite number I…

03:25

Find $d s / d t$ if $s=\left(3 t^{2}+1\right)^{2}\left(t^{3}-6 t+1\right)^{7…

01:39

Let $f(x)=x^{-2 / 3}$ on the interval $[-1,8] .$ Does $f$ have extrema on th…

03:00

A right triangle is formed in the first quadrant by a line passing through t…

03:45

Let $f(x)=3 x^{4}-7 x+10 .$ Approximate $f(1.01)$ by linearizing $f$.(a)…

01:55

Using long division, find the oblique asymptotes.$$f(x)=\left(2 x^{2}+x-…

05:48

(a) Determine the $x$ -intercepts, (b) the vertical asymptotes, (c) the hori…

07:47

A small bottling company finds that it costs $6,000$ to prepare 10,000 six p…

03:13

Differentiate $\left(2 x+x^{4}\right)^{2}$ with respect to $x^{3} .$ Hint: l…

01:17

Use your calculator to compute the expression given.$$4.753^{4 / 3}$$