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Determine where the function is concave upward and downward, and list all inflection points.$$f(x)=\frac{x}{\sqrt{x-4}}$$

$$\mathrm{CU} \text { if } x<16, \text { otherwise } \mathrm{CD} \mathrm{I}\left(16, \frac{8 \sqrt{3}}{3}\right)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Campbell 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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Determine where the functi…

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we want to identify the con cavity and any inflection points of F of X equals X over square root of x minus four. To do so the first time briancon cavity and inflection points. So remember that an inflection point means that double prime changes sign. The sign of a federal crime is what tells us the con cavity. So when we see a change in the cavity, we see an inflection point. Thus, we have to find employment double prime F prime of x minus 8/2 minus two times X minus 4 to 3. Halfs F prime of x minus 16/16 X minus 45 half. So you see we have double prime can change sign. We have to partition F double prime with its critical and partition points for the numerator. We have critical 450.16. The denominator we have partition four. Thus we have to check to the left and right before 16. Since it is undefined for X less than four, we actually only have to check between four and 16 and greater than 16, or rather so after the prime for X between four and 16 technical standards negative for X greater than 16, such as x equals 20 positive. That means FS concave up on 16 infinity, concave down on 4- 16 and has an inflection point at x equals 16.

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