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Determine the extrema on the given interval.$$f(x)=2 x^{3}+3 x^{2}-12 x-6 \text { on: }(a)[-3,2] ;(b)[-5,3]$$

(a) $\mathrm{M}=14, \mathrm{m}=-13$(b) $\mathrm{M}=39, \mathrm{m}=-121$

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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we first want to find the critical points of this function. We take derivatives to this function which is six X squared minus six yards plus six acts sorry minus job. And we can factor it into two parts. Sure X plus two and ex miners one. Then they like the derivative equals zero. And then we got through critical numbers which are actually cause miners too. And nancy goes what? In part A. We want to find the extreme between the interval minus three and two. I know that both of the critical numbers are visiting this interval and hence they varied their function values. We can spray 14 minus 13 and minus two respectively. So 14 is a maximum and minor. 13 is a minimal for part B. We want to find the extreme between the interval minus five minus three and the two critical numbers are still within the syndrome guns. N. We varied the function value as these four points. They're functional values and miners and 121 for change 13 concerning my respective So my nurse 100 and 21 is a minimum and 39 is a maximum

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