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Determine where the function is concave upward and downward, and list all inflection points.$$f(x)=a x^{2}+b x+c:(a) a>0 ; \text { (b) } a<0$$

(a) $\mathrm{CU}$(b) CD

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Missouri State University

Harvey Mudd College

Baylor University

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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Question 24 wants you to find where the function F of X equals X squared, plus BX plus C is concave up or down and list any inflection points. Uh, pro for part A. They're telling you that a is greater than zero. So, uh, to find con cavity, we need the second derivative. So F prime of X is equal to two. A X plus B f double prime of X is equal to two A. This is greater than zero, the whole thing. Therefore, that means we are concave up. And for there to be inflection point after Prime FX would need to be zero. But because this is a constant, it's never gonna be zero. Therefore, where concave up, always part B. They're telling us that a is less than zero. So looking at our function here again after the prime of X is equal to two A. This whole function, though two times a negative number, is a negative number. So this is negative, which means it's lesson zero so or concave down again. There's no place where F double prime of X is equal to zero. Therefore, we're concave down always, and those are your answers to question 24

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