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Determine the extrema of the function defined by the equation $f(x)=\frac{x}{x^{2}+1}$ on $(-\infty, 0] .$ Justify your conclusions.

$$\mathbf{M}=0, \mathbf{m}=-1 / 2$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 1

Extrema of a Function

Derivatives

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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.

05:56

Find the absolute extrema …

05:33

02:49

02:27

Find the indicated extremu…

05:19

03:22

first we round you'll find the critical point of this function. We take the rabbit house to this function and according to the formula of the derivatives of the cautions, the denominator is just as a square. The numerator articles. Yeah, derivative of the numerator. Richard one times the denominator miners. The numerator times the derivative of the denominator. And we can rewrite it as miners adds minus one and adds plus one. The latter derivative equals zero. And we get to critical numbers hr minus one and one not that only one has been the interval from minors, sorry, only minus one is within the injury from miners and vanity to their own. Yeah. And of X goes to minus infinity. Its function value goes to there. The function value at the miners one as minors. One half a functionary of the steroids there. Thus the minimum as miners. Uh huh. And the maximum zero.

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