Homework SourseWe are finding intervals of concavity of fx x4 4x3 18x2 T
We are finding intervals of concavity of f(x) = x4 - 4x3 - 18x2
The possible inflection point list is {-1,3}
Finally, determine the concavity in each interval between possible inflection points. In each interval, choose a test point and determine whether the second derivative of f is positive or negative throughout the interval.
Remember, we know f \'\'(x) = 12x2 - 24x - 36.
Interval Test point a Test f \'\'(a) Concavity
(- infinity,-1) f \'\' > 0, so f conc up
f \'\' < 0, so f conc down
(-1,3) f \'\' > 0, so f conc up
f \'\' < 0, so f conc down
(3, infinity) f \'\' > 0, so f conc up
f \'\' < 0, so f conc down
The possible inflection point list is {-1,3}
Finally, determine the concavity in each interval between possible inflection points. In each interval, choose a test point and determine whether the second derivative of f is positive or negative throughout the interval.
Remember, we know f \'\'(x) = 12x2 - 24x - 36.
Interval Test point a Test f \'\'(a) Concavity
(- infinity,-1) f \'\' > 0, so f conc up
f \'\' < 0, so f conc down
(-1,3) f \'\' > 0, so f conc up
f \'\' < 0, so f conc down
(3, infinity) f \'\' > 0, so f conc up
f \'\' < 0, so f conc down
Solution
You may choose any points in the ranges I would suggest round easy one such as -2 for the first interval 0 for the second and 4 for the third.
