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Concavity (Posted on 2014-08-26)

For the function below, determine the range of x-values for which the function is concave up:

1) Assuming k is negative.
2) Assuming k is positive.

No Solution Yet Submitted by Dustin

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Solution

Comment 2 of 2 |

y = - (x² + 1/k)^-1

Differentiating:     y’      = 2x(x² + 1/k)^-2

and again:             y’’     = 2(x² + 1/k)^-2 – 4x(x² + 1/k)^-32x

                                    = 2(x² + 1/k)^-3(x² + 1/k – 4x²)

                                    = 2(x² + 1/k)^-3(1/k – 3x²)

(1) For k < 0: 1/k – 3x² < 0, so the function will be concave upwards

                        (i.e. y’’ > 0) iff x² + 1/k < 0, giving |x| < sqrt(-1/k).

            i.e. between the asymptotes, x = +/-sqrt(-1/k), as Charlie found,

            with a local minimum at (0, -k), above the origin.

(2) For k>0: (x² + 1/k)^-3 > 0, so the function will be concave upwards

                        where   1/k – 3x² > 0, giving |x| < 1/sqrt(3k).

            i.e. between the points of inflection at (+/-1/sqrt(3k), -3k/4),

            with a local minimum at (0, -k), below the origin.

Posted by Harry on 2014-08-27 13:00:37

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