33b 2 Consider the function For this function there are two

3-3b #2

Consider the function. For this function there are two important intervals:andwhereis a critical number. Find For each of the following intervals, tell whetheris increasing (type in INC) or decreasing (type in DEC). : : For each of the following intervals, tell whetheris concave up (type in CU) or concave down (type in CD). : :

Solution

f(x) = 4(x-5)^2/3
f\'(x) = 8/3 (x-5)^-1/3
for critical point, f\'(x) = 0
8/3 (x-5)^-1/3 = 0
x R except x = 5
hence A = 5
for increasing f\'(x) > 0
hence x > 5
i.e. x (5, )
for decreasing f\'(x) < 0
hence x < 5
i.e. x (-, 5)
f\'\'(x) = -8/9(x-5)^-4/3
for concave up, f\'\'(x) > 0
hence x < 5
i.e. x (-, 5)
for concave down, f\'\'(x) < 0
hence x > 5
i.e. x (5, )