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Determine the extrema on the given interval.$$f(x)=x^{2}-2 x+3 \text { on: }(a)[0,2] ;(b)[2,3] ;(c)(2,3]$$

(a) $\mathrm{M}=3, \mathrm{m}=2$(b) $\mathrm{M}=6, \mathrm{m}=3$(c) $\mathrm{M}=6,$ no min

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 1

Extrema of a Function

Derivatives

Missouri State University

Campbell University

Harvey Mudd College

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.

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first we want to find are critical points of this function. We take the rabbit ears to this function rituals to eggs, miners too. Can we love the derivative equals zero hands. We get a critical number which is one for part A. We want to find the extreme between the interval from 0 to 2 and the critical point has been this interval. Next we buried the function value on the boundary of the interval as well as a critical point. There are 32 and three and enhance. Three is a maximum and two is a minimum for barbie. We understand the extreme between the injectable to. Yeah sort of the critical point is not in this in general. Hence we just evaluate their function value at the boundary of the interval. Have two equals three and F. Three or six. Can't businesses many millions issues in Mexico for percy. It is still the interval from 2 to 3 but two is exclusive. And here are you see red yellow to illustrate this and the function value is three and six. Since X cannot reach two, the function value cannot reach three and hands there is no minimum, and six is to the maximum.

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