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Determine the extrema on the given interval.$$f(x)=4 x^{3 / 4}+2 \text { on: }(a)[0,16] ;(b)(0,16) ;(c)[0,16)$$

(a) $\mathrm{M}=34, \mathrm{m}=2$(b) none(c) $\mathrm{No} \operatorname{Max}, \mathrm{m}=2$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 1

Extrema of a Function

Derivatives

Campbell University

Harvey Mudd College

Boston College

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 a critical points of this function. We take derivatives to this function Major has three times X. to the Power 1 4th. We can also write their sale spray over extra power one force and the director cannot be there or you can only feel stripped of this And we get a critical number which is zero proper A. We want to find extreme Of the function between the interval zero and 16. Not that here zero is both boundary of the in general and the critical point. Hence we just evaluate their function values if there were goes to and of 16 he goes 34 So truly is a minimum and 34 even maximum for power bi the interval is the same but it's and opening dribble here I use a right arrow to illustrate this and since it's an open interval it cannot original function value. So there's no minimum there is no maximum caesar for party boundary there Roy is inclusive Enhance. It can reach two but cannot reach 34 so it has a minimum past still has no maximum

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