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

Always CD

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Missouri State University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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 23 would like you to determine where f of X equals negative three X squared plus two. Is concave up or down? Uh, and if it has any inflection points, so to do so, we need to find the second derivative. So F prime of X is equal to negative. Six X. If double prime of X is equal to negative six because this is less than zero, that tells us where concave down and the only time there's an inflection point is when I have double Prime Vector is equal to zero. Therefore, we are concave down always because there is no inflection point. That's your answer to question 23.

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