Holidad Blog numero tresss

Concavity

You use the same steps as the first derivative test but you take the 2nd derivative.

1. Take the 1st derivative, set it equal to 0, then solve.

2. Take the 2nd derivative, set it equal to 0, then solve.

3. Check for differentiability or non-differentiability.

4. Set up your intervals.

5. Pick #s on the interval and plug in your 2nd derivative.

6. If it is positive then the function is concave up.

If it is negative then the function is concave down.

7. If there is a change it is a point of inflection.

Example:

Determine the points of inflection of f(x) = x^4 - 4x^3

1. n/a

2. 4x^3 - 12x^2

12x^ - 24x = 0

12x(x-2) = 0

x = 0, 2

3. check :)

4. (-infinity, 0)

positive

concave up

(0, 2)

negative

concave down

(2, infinity)

positive

concave up

Therefore x=0, x=2 are points of inflection.