Sheet 6. Investigation of Functions

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) = e^{13x3−3x−4} f) f (x) = e^3x−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) = e^{23x3−12x2−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) = e^x+ 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: October 8, 2009 1

Faculty of Management  Mathematics  Exercises

Exercise 6.5. Find the points of inection 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

2 Exercise 6.6. Suppose the function f satises the following conditions:

a) f : R → R, f (0) = 5, f (1) = 0, f ³₂
= 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 ∈ ³₂, ∞
, f⁰⁰(x) > 0 ⇔ x ∈ −∞,³₂
, f^{00 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 ¹₂

= 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
= f^{00 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: October 8, 2009 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) = e^x

x + 1

Last update: October 8, 2009 3