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(a) Determine the extrema of the function defined by the equation $f(x)=\frac{x^{2}}{x^{3}+1}$ on $[0, \infty) .$ Justify your conclusions.(b) Does this function have extrema on $(-\infty, 0] ?$

(a) $\mathrm{M}=\frac{\sqrt[3]{4}}{3}, \mathrm{m}=0$(b) none

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.

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01:19

first we want to find the critical point of this function. The first to take the rabbit have to this function. And it's the relative looks like this relax the drive letter because zero an Islamic gaps parks he called zero or access because the pill cubic rode out to. And if the jury battlefield to exist that is the discriminatory call. Zero role X equals minus one enhanced against three critical numbers. For part A. It's the interval from their role to politics. The infinite and owning the critical number the cubic rolled out two is visiting this interval. So they find the function values at zero which is zero and a function value at the cubic rolled out two years cubic root out 4/3. And when X goes to positive infinity its function battle goes to zero enhance the cubic root out forward three as a maximum. And there is a meaning prepare B. It is the interval from negative infinity to zero. Note that X equals minus one is not in the demise of the function. Since the denominator equal zero at minus one. So when x goes to my nurse infinity it's functional value goes to zero. The air's goes to minus one. From the left side of the function value goes to negative infinity and then it goes to minors, funds from the right side of the functional writer goes to polar tone finish and after they're right there. Hence this function has neither minimum nor maximum since it can go through negative infinity or positive infant here, I also draw a graph of this function. You can say that in the interval their role to positive infinity it has a minimum which is zero and a maximum. But between the interval from negative infinity to their role, it's kind of richest negative infinity and positive effect and enhance. It has neither maximum not minimum.

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