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Determine where the function is concave upward and downward, and list all inflection points.$$f(x)=\sqrt{x+1}$$

CD everywhere

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Baylor University

University of Michigan - Ann Arbor

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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Determine where the functi…

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question 30 would like you determine where the function F of X equals square X plus one is concave up or down, um, and then find any inflection points. So to do so, you need the second derivative of F of X f of X can be rewritten as X plus one to the one half. So taking that first derivative, the B F prime of X is equal to one half X plus one to the negative one half. From there at double prime of X would be negative 1/4 x plus one to the negative 3/2, which is equivalent to negative 1/4 X plus one took the 3/2. Ah, there is not gonna be any inflection point. So when FDR brand X is equal to zero, this function never equal zero. But there is a assume tote or a place where this function is undefined, and that would be at X is less or X is equal to negative one. Ah, so from there, then you can test um, the con cavity. Uh, this function actually is also undefined when X is less than negative one. Therefore, we are only gonna be testing when X is greater, the negative one, so F double prime of two is equal to negative 1.29 Therefore, you are a concave down when X is greater than negative one. So, technically, Khan gave down everywhere because otherwise your function is undefined. It looks like that, and that's your answer to question 30.

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