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Determine where the function is concave upward and downward, and list all inflection points.$$f(x)=-2 x^{3}+5 x^{2}+7$$

$$\mathrm{CU} \text { if } x<5 / 6, \mathrm{CD} \text { if } x>5 / 6$$

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

Idaho State University

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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Determine the intervals on…

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Find the intervals on whi…

question 26 would like you to determine where the function F of X equals negative two X cubed plus five X squared plus seven is either concave up or down. Uh, enlist any inflection points. So to do so, you need the second derivative, um so f prime of X is equal to negative. Six X squared plus 10 x Have double prime of X is equal to negative 12 x last 10 uh, so find your inflection points. That's when F double prime of X is equal to zero so equal to negative. 12 X plus 10. Solving for X. That gives you 10/12, which is equal to 5/6. Ah, and then plugging an F 5/6. That's equal to about 9.3 14 So your inflection point is five or six. 9.314 Uh, from there. Then you can find where Khan cavity is up or down, uh, by using F double prime of X and testing, a number that is less than 5/6. So F double prime of zero is equal to 10, which means when X is less than 5/6. We are concave up and testing a number where X is greater than five or six F double prime of 5/3 is equal to negative 10. So when X is greater than 5/6, you are concave down, and those are all your answers to question 26.

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