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(a) Use a graph of $f$ to estimate the maximum and minimum values. Then find the exact values.(b) Estimate the value of $x$ at which $f$ increases most rapidly. Then find the exact value.$f(x)=\frac{x+1}{\sqrt{x^{2}+1}}$

a. local and absolute max at $(1, \sqrt{2})$b. $x=\frac{3-\sqrt{17}}{4} \approx-0.28078$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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for this question. Where you first right. Doubts on the purity of four fayv in the four part A. We said if prime equals zero, that gives us X equals to one So we can see the other for intervals minus infinity to one. And the one to infinity on the first in parole if prime miss positive. So the function is increasing on the second Herbold if prime Yes, negative. The functions de crazy. That means we have Uh huh. We have, Ah, a local maximum at X seacoast, one with value F one. Because the root off to also this is, um, global. Next, Because the functions uniformly increasing only left the functions uniformity. Christian wings. Right. So it has to be a global maximum. And for Poppy, we are looking for him. Um, we're looking for ah, very off aches at which have increased if the function if increases most rapidly s so we can see that if is increasing, F is increasing. Own miners even need toe one. So it take the second of your it If we have two X square minus three x minus one over X square pass one to the 5/2 we said her psychology. Unity to be zero. So we have exit close to £3 of miners fruit off 17 divided by four that will restrict our domain to B minus evenly toe one. So that domain, because this is the increasing terrible And that means we only can see there x to be three minus root of 17/4. So on the left, from minus infinity to the reminders wrote off 17 or four s top of time. Um, it's positive that means f prime. The first of the purity of forfeit off F is increasing on the right. Three minus root off 17 divided by 4 to 1 over this interval if double promise negative. So if crime is decreasing, then use f prime has ah, no coal. Um, maximum at X ecosystem reminders wrote off 70 Divide by four and and this is also a global next for the same earth. So one X equals 23 minus root of 17 divided by four. If prime has the has a maximum value, which means the function f increasing most rapidly at this point

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