33b 1 Consider the function has inflection points at reading

3-3b #1

Consider the function. has inflection points at (reading from left to right),, and whereis andis andis For each of the following intervals, tell whetheris concave up (type in CU) or concave down (type in CD). : : : :

Solution

f(x) = 12x^5 + 45x^4 - 80x^3 + 1
f\'(x) = 60x^4 + 180x^3 - 240x^2
f\'\'(x) = 240x^3 + 540x^2 - 480x
for inflection points, f\'\'(x) = 0
240x^3 + 540x^2 - 480x = 0
60x(4x^2 + 9x - 8) = 0
x = 0, 4x^2 + 9x - 8 =0
x = 0, x = [-9 + 81+128]/8, [-9 - 81+128]/8
x = 0, x = [-9+209]/8, [-9-209]/8 are the inflection points
D = [-9-209]/8
E = 0
F = [-9+209]/8
for concave up f\'\'(x) > 0 & for concave down f\'\'(x) < 0
(-, D) gives concave down
(D,E) gives concave up
(E,F) gives concave down
(F,) gives concave up