You can locate a role'south concavity (where a function is concave up or down) and inflection points (where the concavity switches from positive to negative or vice versa) in a few simple steps. The following method shows yous how to find the intervals of concavity and the inflection points of

image0.png

  1. Notice the second derivative of f.

    image1.png

  2. Set the second derivative equal to nil and solve.

    image2.png

  3. Determine whether the 2d derivative is undefined for any ten-values.

    image3.png

    Steps ii and 3 requite you lot what you could call "2nd derivative critical numbers" of f because they are analogous to the critical numbers of f that you find using the first derivative. But this set of numbers has no special name. In any effect, the important thing to know is that this listing is fabricated up of the zeros of f ′′ plus any x-values where f ′′ is undefined.

  4. Plot these numbers on a number line and examination the regions with the second derivative.

    Apply –2, –1, 1, and 2 as examination numbers.

    image4.png

    Because –2 is in the left-about region on the number line below, and considering the second derivative at –two equals negative 240, that region gets a negative sign in the figure below, and so on for the other 3 regions.

    A second derivative sign graph.

    A second derivative sign graph

    A positive sign on this sign graph tells you that the part is concave up in that interval; a negative sign means concave down. The function has an inflection point (usually) at any x-value where the signs switch from positive to negative or vice versa.

    image6.png

    If you go a problem in which the signs switch at a number where the second derivative is undefined, you accept to check one more affair before last that there's an inflection point there. An inflection bespeak exists at a given 10-value simply if there is a tangent line to the role at that number. This is the case wherever the get-go derivative exists or where in that location's a vertical tangent.

  5. Plug these iii x-values into f to obtain the function values of the 3 inflection points.

    A graph showing inflection points and intervals of concavity.

    A graph showing inflection points and intervals of concavity

    image8.png

    The foursquare root of two equals about 1.4, so in that location are inflection points at about (–1.4, 39.half-dozen), (0, 0), and almost (one.4, –39.6).

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About the volume author:

Steven Holzner was an honour-winning writer of more than 130 books, of which more than than ii million copies have been sold. His books accept been translated into 23 languages. He served on the Physics faculty at Cornell University for more than than a decade, instruction both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate piece of work at Massachusetts Plant of Technology, where he besides served as a faculty member.

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