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

$$\mathrm{CU} \text { if } x \neq 0$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

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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Determine where the functi…

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Find the intervals on whi…

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question 39 would like you to determine when the function f of X equals X minus X to the to over three is concave up or down and list any inflection points. So to do so, you need to find the second derivative of F of X so f prime of X. It's just gonna be one minus 2/3 x to the negative one third F double prime of X then would just be to over nine x to the negative 4/3. From there, that simplifies to two over nine X to the 4/3. This function Ah, to find inflection points you'd want When f double prime of X is equal to zero, it never equal zero, but it is undefined at X equals zero. There's an ASEM tote. Um, therefore, you can test, um, this function when X is less than zero and when X is greater than zero with F double prime of X to see if it's concave up or down. So f double prime of negative two. This 0.8 That means that's concave up and F double prime up to is equal 2.8 as well, so it's also concave up, which means f of X is concave up everywhere if X does not equal zero and that's your answer to question 39.

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