Faculty of Management – Mathematics – Exercises
Sheet 6. Investigation of Functions
Exercise 6.1. Find the asymptotes of the graphs of the given functions:
a) f (x) = 1
1 − x2 b) f (x) = x2
2x + 3 c) f (x) = x x2+ 1 d) f (x) = x3+ x2
x2− 4 e) f (x) = x − 3
√x2− 9 f) f (x) =√
1 + x2+ 2x
g) f (x) =
√1 + x2
x h) f (x) = sin x
x i) f (x) = x2e−x
Exercise 6.2. Find the intervals of increase and decrease of the given functions:
a) f (x) = 2x3− 15x2+ 36x − 14 b) f (x) = x4+ 4x − 2 c) f (x) = x x2+ 4 d) f (x) = x2+ 3x + 16
x + 3 e) f (x) = e13x3−3x−4 f) f (x) = e3x−x3 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:
a) f (x) = x3+ 3x2− 9x + 7 b) f (x) = 1
3x3− 3x2+ 5x + 9 c) f (x) = −x3+ 9x2− 24x + 17 d) f (x) = 3x5+ 5x3− 11 e) f (x) = x4− 4x3+ 4x2− 11 f) f (x) = 13x3− 12x2− 2x + 1 g) f (x) = 2x3− 15x2+ 36x − 14 h) f (x) = x4+ 4x − 2 i) f (x) = e−13x3−12x2+2x−2 k) f (x) = e−13x3−12x2+6x+10 l) f (x) = e23x3−12x2−x+3 m) f (x) = x
x2+ 4 n) f (x) = x2+ 1
x2+ 4 o) f (x) = 3x
x2+ x + 1 p) f (x) = x2− 3x + 4 x − 3 q) f (x) = x2+ x + 9
x + 1 r) f (x) = x2− 2x − 3
x − 2 s) f (x) = 2 + x − x2 x − 1 t) f (x) = −x2+ 2x + 9
x + 2 u) f (x) = 8x − 4x2− 1
2 − x v) f (x) = (1 − x)2 2x w) f (x) = x −√
x 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) = x33 + x2− 3x, x ∈ h−4, 4i f) f (x) = 23x3− x22 − x −13, x ∈ h−2, 2i g) f (x) = −x33 − x2+ 3x + 23, 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 l) f (x) = 2x3− 3x2− 36x − 8, x ∈ h−3, 6i m) f (x) = x − 2√
x, x ∈ h0, 5i n) f (x) = x2ln x, x ∈ h1, ei
Last update: December 2, 2008 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:
a) f (x) = x4− 12x3+ 48x2 b) f (x) = x2 − 5x + 6
x + 1 c) (x) = x + sin 2x d) f (x) = xe−x e) f (x) = ln x
x f) f (x) = x4 12 −x3
3 + x2 Exercise 6.6. Suppose the function f satisfies the following conditions:
a) f : R → R, f (0) = 5, f (1) = 0, f¡3
2
¢ = 12, f (2) = 1, f (3) = 0, lim
x→−∞f (x) = ∞, lim
x→∞f (x) =
−∞,
f0 : R → R, f0(x) > 0 ⇔ x ∈ (1, 2) , f0(x) < 0 ⇔ x ∈ (−∞, 1) ∪ (2, ∞) , f0(1) = f0(2) = 0, f00 : R → R, f00(x) < 0 ⇔ x ∈¡3
2, ∞¢
, f00(x) > 0 ⇔ x ∈¡
−∞,32¢
, f00¡3
2
¢= 0.
b) f : R → R, f (−2) = 0, f (−1) = −2, f (0) = −4, f (1) = 0, lim
x→−∞f (x) = −∞, lim
x→∞f (x) = ∞, f0 : R → R, f0(x) > 0 ⇔ x ∈ (−∞, −2)∪(0, ∞) , f0(x) < 0 ⇔ x ∈ (−2, 0) , f0(−2) = f0(0) = 0, f00 : R → R, f00(−1) = 0, f00(x) > 0 ⇔ x ∈ (−1, ∞) , f00(x) < 0 ⇔ x ∈ (−∞, −1) .
c) f : R → R, f (−1) = 0, f ¡
−12¢
= 1, f (0) = 2, f¡1
2
¢ = 1, f (1) = 0, lim
x→−∞f (x) = ∞,
x→∞lim f (x) = ∞,
f0 : R → R, f0(x) < 0 ⇔ x ∈ (−∞, −1) ∪ (0, 1) , f0(x) > 0 ⇔ x ∈ (−1, 0) ∪ (1, ∞) , f0(−1) = f0(0) = f0(1) = 0,
f00 : R → R, f00¡
−12¢
= f00¡1
2
¢ = 0, f00(x) < 0 ⇔ x ∈ ¡
−12,12¢
, f00(x) > 0 ⇔ x ∈ ¡
−∞, −12¢
¡1 ∪
2, ∞¢ .
d) f : R → R, f (−4) = −12, f (−2) = −2, f (0) = 0, f (2) = 2, f (4) = 12, lim
x→−∞f (x) = 0,
x→∞lim f (x) = 0,
f0 : R → R, f0(x) < 0 ⇔ x ∈ (−∞, −2)∪(2, ∞) , f0(x) > 0 ⇔ x ∈ (−2, 2) , f0(−2) = f0(2) = 0, f00 : R → R, f00(−4) = f00(4) = 0, f00(x) > 0 ⇔ x ∈ (−4, 0) ∪ (4, ∞) , f00(x) < 0 ⇔ x ∈ (−∞, −4) ∪ (0, 4) .
e) f : R\ {−1, 1} → R, f (0) = 0, lim
x→−∞f (x) = 0, lim
x→∞f (x) = 0, lim
x→−1−f (x) = ∞, lim
x→−1+f (x) =
−∞, lim
x→1−f (x) = ∞, lim
x→1+f (x) = −∞,
f0 : R\ {−1, 1} → R, f0(x) > 0 ⇔ x ∈ R\ {−1, 1} ,
f00 : R\ {−1, 1} → R, f00(x) > 0 ⇔ x ∈ (−∞, −1) ∪ (0, 1) , f00(x) < 0 ⇔ x ∈ (−1, 0) ∪ (1, ∞) . f) f : R\ {−2, 2} → R, f (0) = 1, f (−1) = f (1) = 0 lim
x→−∞f (x) = 2, lim
x→∞f (x) = 2, lim
x→−2−f (x) =
∞, lim
x→−2+f (x) = −∞, lim
x→2−f (x) = −∞, lim
x→2+f (x) = ∞,
f0 : R\ {−2, 2} → R, f0(0) = 0, f0(x) > 0 ⇔ x ∈ (−∞, −2) ∪ (−2, 0) , f0(x) < 0 ⇔ x ∈ (0, 2) ∪ (2, ∞) ,
f00 : R\ {−2, 2} → R, f00(x) > 0 ⇔ x ∈ (−∞, −2) ∪ (2, ∞) , f00(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) c) f (x) = x 1 − x2 d) f (x) = x3
x − 1 e) f (x) = x√
1 − x2 f) f (x) =√ x − x g) f (x) = ln x
x h) f (x) = e−x2 i) f (x) = ex
x + 1
Last update: December 2, 2008 3
