Byron S. answered • 11/16/14

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Hi Alison,

To determine the concavity of a function, you need to know the sign of the 2nd derivative over the particular intervals between inflection points and discontinuities. This particular function has no discontinuities, so you only have to worry about inflection points as boundaries.

If f'' is positive, f is concave up. If f'' is negative, f is concave down.

Given inflection points at x ≈ 4 and -4, you get three regions to test:

(-∞. -4), (-4, 4), (4, +∞)

In each case, you just need to plug in a value and determine what the sign will be. You don't have to evaluate the function fully.

f"(x) = -(294 (3x^2 - 49)) / (x^2 +49)^3

The factor 294 won't change the sign, so you can ignore it. The denominator is always positive, so that has no effect on the sign either. What's left is -(3x^2 - 49)

For the first interval, try plugging in -5 (or -6 or -7).

-(3x^2 - 49) @x=-5

= -(3(-5)^2 - 49)

= -(75-49)

This is a negative number, so on the interval (-∞. -4), f(x) is concave down.

For the second interval, x=0 is easy to plug in:

-(3x^2 - 49) @x=0

= -(3(0)^2 - 49)

= +49

On the interval (-4, 4), f(x) is concave up.

For the third interval, test x=5 (or 6 or 7), and you'll get the same results at for -5. f(x) is concave down on (4, +∞).