Faculty of Management – Mathematics – Exercises
Sheet 6. Investigation of Functions
Exercise 6.1. Find the asymptotes of the graphs of the given functions:
1
x2
x
a) f (x) =
b)
f
(x)
=
c) f (x) = 2
2
1−x
2x + 3
x +1
3
2
√
x +x
x−3
e) f (x) = √
d) f (x) = 2
f) f (x) = 1 + x2 + 2x
x −4
x2 − 9
√
1 + x2
sin x
g) f (x) =
h) f (x) =
i) f (x) = x2 e−x
x
x
Exercise 6.2. Find the intervals of increase and decrease of the given functions:
x
a) f (x) = 2x3 − 15x2 + 36x − 14 b) f (x) = x4 + 4x − 2
c) f (x) = 2
x +4
2
x + 3x + 16
1 3
3
d) f (x) =
e) f (x) = e 3 x −3x−4
f) f (x) = e3x−x
x+3
g) f (x) = xe−3x
h) f (x) = x − ln(1 + x) i) f (x) = (x2 − 3) e−x
Exercise 6.3. Find the local maximum and minimum values of the following functions:
1
a) f (x) = x3 + 3x2 − 9x + 7
b) f (x) = x3 − 3x2 + 5x + 9 c) f (x) = −x3 + 9x2 − 24x + 17
3
d) f (x) = 3x5 + 5x3 − 11
e) f (x) = x4 − 4x3 + 4x2 − 11
1 3
1 3 1 2
− 2 x +6x+10
x2 + 1
x2 + 4
x2 + x + 9
q) f (x) =
x+1
x2 + 2x + 9
t) f (x) = −
x+2
√
w) f (x) = x − x
n) f (x) =
2 3 1 2
− 2 x −x+3
l) f (x) = e 3 x
3x
+x+1
2
x − 2x − 3
r) f (x) =
x−2
8x − 4x2 − 1
u) f (x) =
2−x
o) f (x) =
1 2
i) f (x) = e− 3 x − 2 x +2x−2
x
m) f (x) = 2
x +4
x2 − 3x + 4
p) f (x) =
x−3
2 + x − x2
s) f (x) =
x−1
(1 − x)2
v) f (x) =
2x
g) f (x) = 2x3 − 15x2 + 36x − 14 h) f (x) = x4 + 4x − 2
k) f (x) = e− 3 x
f) f (x) = 13 x3 − 12 x2 − 2x + 1
x2
x) f (x) = ex + e−x
Exercise 6.4. Find the absolute maximum and absolute minimum of a function f on the given
interval:
a) f (x) = x4 − 32x, x ∈ h−2, 3i
b) f (x) = x3 − 3x2 + 6x − 5, x ∈ h−1, 1i
c) f (x) = 3x − x3 , x ∈ h0, 4i
d) f (x) = x4 − x2 , x ∈ h−1, 2i
e) f (x) =
x3
3
+ x2 − 3x, x ∈ h−4, 4i
3
f) f (x) = 23 x3 −
x2
2
− x − 31 , x ∈ h−2, 2i
g) f (x) = − x3 − x2 + 3x + 32 , x ∈ h−4, 2i
h) f (x) = x3 − 3x2 + 3x + 1, x ∈ h0, 3i
i) f (x) = −2x3 + 3x2 − 6x − 2, x ∈ h−1, 2i
j) f (x) = x3 − 3x2 + 15x − 3, x ∈ h−1, 1i
k) f (x) = x2 − 2x + 3, x ∈ h−2, 5i
√
m) f (x) = x − 2 x, x ∈ h0, 5i
l) f (x) = 2x3 − 3x2 − 36x − 8, x ∈ h−3, 6i
Last update: December 2, 2008
n) f (x) = x2 ln x, x ∈ h1, ei
1
Faculty of Management – Mathematics – Exercises
Exercise 6.5. Find the points of inflection and the intervals on which the given functions are convex
and concave:
x2 − 5x + 6
4
3
2
a) f (x) = x − 12x + 48x b) f (x) =
c) (x) = x + sin 2x
x+1
ln x
x4 x3
d) f (x) = xe−x
e) f (x) =
f) f (x) =
−
+ x2
x
12
3
Exercise 6.6. Suppose the function f satisfies the following conditions:
¡ ¢
a) f : R → R, f (0) = 5, f (1) = 0, f 32 = 12 , f (2) = 1, f (3) = 0, lim f (x) = ∞, lim f (x) =
x→−∞
x→∞
−∞,
f 0 : R → R, f 0 (x) > 0 ⇔ x ∈ (1,
2) , ¢f 0 (x) < 0 ⇔ x ∈ (−∞,
1) ∪ ¢(2, ∞)¡ , ¢f 0 (1) = f 0 (2) = 0,
¡
¡
f 00 : R → R, f 00 (x) < 0 ⇔ x ∈ 23 , ∞ , f 00 (x) > 0 ⇔ x ∈ −∞, 23 , f 00 23 = 0.
b) f : R → R, f (−2) = 0, f (−1) = −2, f (0) = −4, f (1) = 0, lim f (x) = −∞, lim f (x) = ∞,
x→−∞
x→∞
f 0 : R → R, f 0 (x) > 0 ⇔ x ∈ (−∞, −2)∪(0, ∞) , f 0 (x) < 0 ⇔ x ∈ (−2, 0) , f 0 (−2) = f 0 (0) = 0,
f 00 : R → R, f 00 (−1) = 0, f 00 (x) > 0 ⇔ x ∈ (−1, ∞) , f 00 (x) < 0 ⇔ x ∈ (−∞, −1) .
¡ ¢
¡ ¢
c) f : R → R, f (−1) = 0, f − 12 = 1, f (0) = 2, f 12 = 1, f (1) = 0, lim f (x) = ∞,
x→−∞
lim f (x) = ∞,
x→∞
0
f : R → R, f 0 (x) < 0 ⇔ x ∈ (−∞, −1) ∪ (0, 1) , f 0 (x) > 0 ⇔ x ∈ (−1, 0) ∪ (1, ∞) ,
f 0 (−1) = f 0 (0) = f 0 (1) = 0,
¡
¢
¡
¢
¡ ¢
¡ ¢
f¡ 00 : R¢ → R, f 00 − 12 = f 00 12 = 0, f 00 (x) < 0 ⇔ x ∈ − 21 , 21 , f 00 (x) > 0 ⇔ x ∈ −∞, − 21 ∪
1
,∞ .
2
d) f : R → R, f (−4) = − 12 , f (−2) = −2, f (0) = 0, f (2) = 2, f (4) =
1
, lim f
2 x→−∞
(x) = 0,
lim f (x) = 0,
x→∞
0
f : R → R, f 0 (x) < 0 ⇔ x ∈ (−∞, −2)∪(2, ∞) , f 0 (x) > 0 ⇔ x ∈ (−2, 2) , f 0 (−2) = f 0 (2) = 0,
f 00 : R → R, f 00 (−4) = f 00 (4) = 0, f 00 (x) > 0 ⇔ x ∈ (−4, 0) ∪ (4, ∞) , f 00 (x) < 0 ⇔ x ∈
(−∞, −4) ∪ (0, 4) .
e) f : R\ {−1, 1} → R, f (0) = 0, lim f (x) = 0, lim f (x) = 0, lim − f (x) = ∞, lim + f (x) =
x→−∞
x→∞
x→−1
x→−1
−∞, lim− f (x) = ∞, lim+ f (x) = −∞,
x→1
x→1
f 0 : R\ {−1, 1} → R, f 0 (x) > 0 ⇔ x ∈ R\ {−1, 1} ,
f 00 : R\ {−1, 1} → R, f 00 (x) > 0 ⇔ x ∈ (−∞, −1) ∪ (0, 1) , f 00 (x) < 0 ⇔ x ∈ (−1, 0) ∪ (1, ∞) .
f) f : R\ {−2, 2} → R, f (0) = 1, f (−1) = f (1) = 0 lim f (x) = 2, lim f (x) = 2, lim − f (x) =
x→−∞
x→∞
x→−2
∞, lim + f (x) = −∞, lim− f (x) = −∞, lim+ f (x) = ∞,
x→−2
x→2
x→2
f 0 : R\ {−2, 2} → R, f 0 (0) = 0, f 0 (x) > 0 ⇔ x ∈ (−∞, −2) ∪ (−2, 0) , f 0 (x) < 0 ⇔ x ∈
(0, 2) ∪ (2, ∞) ,
f 00 : R\ {−2, 2} → R, f 00 (x) > 0 ⇔ x ∈ (−∞, −2) ∪ (2, ∞) , f 00 (x) < 0 ⇔ x ∈ (−2, 2) .
Sketch the graph of f.
Last update: December 2, 2008
2
Faculty of Management – Mathematics – Exercises
Exercise 6.7. Investigate the function f and then sketch its graph:
a) f (x) = x3 − 3x2 + 4
b) f (x) = (x − 1)2 (x + 2)
x3
x−1
ln x
g) f (x) =
x
√
e) f (x) = x 1 − x2
d) f (x) =
Last update: December 2, 2008
−x2
h) f (x) = e
x
1 − x2
√
f) f (x) = x − x
c) f (x) =
ex
i) f (x) =
x+1
3