c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
Math 151 Week in Review 11
Sections 5.1, 5.2, & 5.3
1. Given the graph of f ′ below, find the following. Then sketch a graph of f if f is continuous and
f (−6) = 0.
(i) The intervals where f is increasing and decreasing.
(ii) The x-values of any local extrema of f .
(iii) The intervals where f is concave up and concave down.
(iv) The x-values of any inflection points of f .
2. Sketch a graph of a function f with the following properties:
(i) lim f (x) = ∞, lim f (x) = −∞
x→3−
x→3+
(ii) lim f (x) = 0
x→±∞
(iii) f is continuous everywhere except at x = 3.
(iv) f (0) = −3, f (7) = 3
(v) f ′ (x) > 0 on (−2, 3) ∪ (3, 7)
(vi) f ′ (x) < 0 on (−∞, −2) ∪ (7, ∞)
(vii) f ′′ (x) > 0 on (−5, 3) ∪ (5, 7) ∪ (7, ∞)
(viii) f ′′ (x) < 0 on (−∞, −5) ∪ (3, 5)
1
c
Math 151 WIR, Spring 2010, Benjamin
Aurispa
3. Find all critical values for the following functions.
(a) f (x) = x3 − 7x2 + 4
1
(b) f (x) = x 3 (x − 1)2
ex
(c) f (x) =
x+5
(d) f (x) = |x2 − 5x|
4. Find all absolute and local extrema for the following functions by graphing.
(a) f (x) = x2 − 3, −1 ≤ x < 2
(b) f (x) =
(
x+2
−(x − 1)2
if − 2 < x ≤ 0
if 0 < x < 3
5. Find the absolute maximum and minimum values for the following functions on the given interval.
(a)
(b)
(c)
(d)
f (x) = x4 − 8x2 + 1 on [−1, 3]
f (x) = x + 2 sin x on [0, π]
√
f (x) = 3 x2 − 6x on [−1, 4]
1
on [1, 3]; on (3, 4]
f (x) = x−2
6. Sketch a general graph with the following properties.
(a) x = 3 is a critical number of f , but f has no local extrema.
(b) f is continuous and has a local minimum at x = 3, but f ′ (3) does not exist.
7. Find the value of c that satisfies the Mean Value Theorem for the function f (x) = 3 − x2 on the
interval [1, 6].
8. Given that f (2) = 5 and that −4 ≤ f ′ (x) ≤ 4 for all x in the interval [−1, 2], what are the largest and
smallest values of f (−1)?
9. Find the intervals where the following functions are increasing/decreasing and identify all local extrema.
2
(a) f (x) = xe−2x
x+3
(b) f (x) =
(x − 2)2
(c) f (x) = (x − π2 ) cos x − sin x on [0, 2π]
10. Find the intervals where the following functions are concave up and concave down and identify all
inflection points.
(a) f (x) = ex (x3 − 8x2 + 27x − 38)
(b) f (x) = x ln(x − 3)
11. Determine all intervals where the function f (x) = x6 − 6x5 is increasing and decreasing, concave up
and concave down. Identify all local extrema and points of inflection. Then, sketch a graph.
12. Given the following information, identify any local extrema.
x f (x) f ′ (x) f ′′ (x)
−3
1
0
−4
4
5
0
0
6
3
0
12
8
4 DNE DNE
2