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Determine where the function is concave upward and downward, and list all inflection points.$$f(x)=\sqrt{9+x^{2}}$$

Always CU

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

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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question 36 would like you determine where the function is concave up or down, Um, and for the function F of X equals skirt nine plus X squared and then also list any inflection points. So first, what you need to do for FX equals square root nine plus X squared. You need to find the second derivative so first taking F prime of X. This is equal to nine plus X squared to the one half. So just taking the dirt of that you have one half two x because of the chain roll nine plus X squared to the negative one half, that is equivalent two X divided by nine plus X squared to the one half. Now using the quotient rule where F is equal to x f prime is one g is nine plus x squared to the one half, and G prime is equal to one half times two x nine plus x squared to the negative one half. Yeah. From there we can plug that in to our quotient role and get nine plus x square to the one half minus x times just X. Because the one happened to cancel out times nine plus X squared to the one half or the negative one half, all divided by nine plus X squared. Because nine plus x squared to the one half square, it is just nine plus X squared from their F double prime of X. Continuing to just simplify is if you divide each of these by nine plus X squared, you get one over nine plus X squared to the one half minus X squared, divided by nine plus X squared 2 3/2. Multiply this side by nine plus X squared, divided by nine plus X squared. So just multiplying by one. Technically, you would have F double Prime of X is equal to nine plus X squared minus X squared, divided by nine plus X squared 3/2 and I have a common denominator. So this is just nine divided by nine plus X squared to the 3/2. From there, ah, you would have an inflection point when F prime of X is equal to zero. In this case, that does not occur. Um, and this function is always defined. Therefore, we can just test anywhere to see if it's concave up or down. so testing f double prime of zero That's equal 2.333 which means that we are concave up everywhere, and that is your answer to question 36.

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