MAT 210Practice – Second Derivative, Concavity
1)If we find the critical points from the first derivative of a function, what may they represent?
2)If we find the critical points from the second derivative of a function, what may they represent?
3) Let f(x) = –x3 + 9x2 – 24x + 6.
a) Find the intervals in which f(x) is increasing.
b)Find the intervals in which f(x) is decreasing.
c) Find all local extrema.
d) Find all intervals in which f(x) is concave up.
e) Find all intervals in which f(x) is concave down.
f) Find all points of inflection.
4) Let f(x) = x3 – x2 – 21x + 15.
a) Find the intervals in which f(x) is increasing.
b)Find the intervals in which f(x) is decreasing.
c) Find all local extrema.
d) Find all intervals in which f(x) is concave up.
e) Find all intervals in which f(x) is concave down.
f) Find all points of inflection.
5) Let f(x) = x2 – 2x + 3.
a) Find the intervals in which f(x) is increasing.
b)Find the intervals in which f(x) is decreasing.
c) Find all local extrema.
d) Find all intervals in which f(x) is concave up.
e) Find all intervals in which f(x) is concave down.
f) Find all points of inflection.
6) Let f(x) =
a) Find the intervals in which f(x) is increasing.
b)Find the intervals in which f(x) is decreasing.
c) Find all local extrema.
d) Find all intervals in which f(x) is concave up.
e) Find all intervals in which f(x) is concave down.
f) Find all points of inflection.
Answers (Revision 1)
1) The location(s) of local extrema (max’s & min’s)
2) The location of inflection points
3a)
b) and
c) Local Minimum at (2, -14) and Local Maximum at (4, -10)
d)
e)
f) (3, -12)
4a) andb)
c) Local Minimum at (3, -30) and Local Maximum at
d) e) f)
5a)
b)
c) Absolute Minimum at (1, 2)
d)
e) none
f) none
6a)
b)
c) Absolute Maximum at (0, 1)
d)and
e)
f)and