Calculus
posted by Shalma .
Suppose f is continuous on [0, 6] and satisfies the following
x 0 3 5 6
f 1 4 1 3
f' 5 0 8 0
f" 1 3 DNE 3
x 0<x<3 3<x<5 5<x<6
f' +  
f"   +
(they are both tables)
Identify the points of inflection
a. There are no points of inflection
b. (5, 1) only
c. (3,4) and (6, 3)
d. (3,4) only
e. (3,4),(6,3), and (5, 1)
Please explain in details how you got the answer
THANK YOU :)

Answer is b. (5,1) only
Explanation:
Points of inflection are present when f"=0 or is undefined (DNE). From the first table, you can see that f" is not 0 for any values of x, but is undefined for x=5, so there can be a possible point of inflection at x=5. Then, look at the second table. Points of inflection are where concavity changes, where f" changes from one sign to another ( to +, or + to ). At x=5, the only possible point of inflection taken from the first table, f" does change from  to +, so there is a point of inflection at x=5. At x=5, f=1, so the point is (5,1) (aka the answer is b).