[graphing part 2][section 15]

(1.) y = x² - x + 30 here is the problem

y' = 2x - 1 take the derivative

2x - 1 = 0 set the derivative equal to 0

+ 1 +1 add 1 to each side
_______________
2x = 1 add
____ _____
2 2 divide each side by 2

x = 1/2 cancel

y(1/2) = (1/2)² - (1/2) + 30 replace x with 1/2

y(1/2) = 29.75 combine like terms

result: (1/2, 29/75) is the extreme point

y" = 2 take the derivative

result: The graph is concave up for all real numbers, x.

(2.) y = x³ - 12x + 10 here is the problem

y' = 3x² - 12 take the derivative

3x² - 12 = 0 set the derivative equal to 0

+ 12 +12 add 12 to each side
_________________
3x² = 12 add
___ ___
3 3 divide each side by 3

x² = 4 divide and cancel

x = 2 , x = -2 take square roots

y(2) = (2)³ - 12(2) + 10 y(-2) = (-2)³ - 12(-2) + 10

[replace x with 2][replace x with -2]

y(2) = -6 y(-2) = 26 multiply

[and combine like terms]

results: (2,-6) and (-2,26) are the extreme points

y" = 6x take the 2nd derivative

6x = 0 set the 2nd derivative equal to 0
___ ___
6 6 divide each side by 6

x = 0 divide and cancel

result: The point of inflection is (0, 10)

y"(2) = 6(2) y"(-2) = 6(-2) replace x with 2 & -2

[in the 2nd derivative]

y"(2) = 12 y"(-2) = -12 multiply

results: (2,-6) is a MINIMUM & (-2, 26) is a MAXIMUM

(3.) y = x³ - 3x² - 45x + 25 here is the problem

y' = 3x² - 6x - 45 take the derivative

3x² - 6x - 45 = 0 set the derivative equal to 0
____ ____ ___ ___
3 3 3 3 divide thru by 3

x² - 2x - 15 = 0 divide and cancel

(x - 5)(x + 3) = 0 factor

x - 5 = 0 x + 3 = 0 set each factor equal to 0

+ 5 +5 - 3 -3 add this to each side
_____________ ____________
x = 5 , x = -3 add

y(5) = (5)³ - 3(5)² - 45(5) + 25 replace x with 5

y(-3) = (-3)³ - 3(-3)² - 45(-3) + 25 replace x with -3

y(5) = 125 - 75 - 225 + 25 multiply

y(-3) = -27 - 27 + 135 + 25 multiply

y(5) = -150 y(-3) = 106 combine like terms

results: (5, -150) and (-3, 106) are extreme points

y" = 6x - 6 take the 2nd derivative

6x - 6 = 0 set the 2nd derivative equal to 0

+ 6 + 6 add 6 to each side
_____________
6x = 6 add
___ ____
6 6 divide each side by 6

x = 1 divide and cancel

y(1) = (1)³ - 3(1)² - 45(1) + 25 replace x with 1

y(1) = -22 multiply combine like terms

result: (1, -22) is the point of inflection

y"(5) = 6(5) - 6 y"(-3) = 6(-3) - 6

[replace x with 5 in y"][replace x with -2 in y"]

y"(5) = 24 y"(-3) = -24 multiply combine like terms

results: (5, -150) is a MINIMUM point

and (-3, 106) is a MAXIMUM point

(4.) y = 4x³ - 3x + 2

y' = 12x² - 3 take the derivative

12x² - 3 = 0 set the derivative equal to 0

+ 3 + 3 add 3 to each side
_____________
12x² = 3 add
____ _____
12 12 divide each side by 12

x² = 1/4 reduce and cancel

x = 1/2 x = -1/2 take square roots

y(1/2) = 4(1/2)³ - 3(1/2) + 2 y(-1/2) = 4(-1/2)³ - 3(-1/2) + 2

[replace x with 1/2 and with -1/2]

y(1/2) = (1/2) - (3/2) + 2 y(-1/2) = (-1/2) + (3/2) + 2

[multiply]

y(1/2) = 1 y(-1/2) = 3 combine like terms

results: (1/2, 1) and (-1/2, 3) are extreme points

y" = 24x take the 2nd derivative

24x = 0 set the 2nd derivative equal to 0
____ ___
24 24 divide each side by 24

x = 0 divide and cancel

result: (0, 2) is the point of inflection

y" = 24(1/2) y" = 24(-1/2) replace x with 1/2 & -1/2 in y"

y" = 12 y" = -12 multiply

results: (1/2, 1) is a minimum (-1/2, 3) is a maximum


(5.) y = 2x³ here is the problem

y' = 6x² take the derivative

6x² = 0 set the derivative equal to 0
____ _____
6 6 divide each side by 6

x² = 0 divide and cancel

x = 0 take square roots

y = 6(0)³ replace x with 0

y = 0 multiply

result: (0,0) is a critical point

y" = 12x take the 2nd derivative

12x = 0 set the 2nd derivative equal to 0
____ ____
12 12 divide each side by 12

x = 0 divide and cancel

result: The point of inflection is (0,0).

(6.) y = (1/3)x³ + (1/2)x² - 2x - (2/3)
y' = x² + x - 2 take the derivative

x² + x - 2 = 0 set the derivative equal to 0

(x + 2)(x - 1) = 0 factor

x + 2 = 0 x - 1 = 0 set each factor equal to 0

-2 -2 + 1 +1 add this to each side
___________ ______________
x = -2 , x = 1 add

y = (1/3)(-2)³ + (1/2)(-2)² - 2(-2) - (2/3) replace x with -2

y = (1/3)(1)³ + (1/2)(1)² - 2(1) - (2/3) replace x with 1

y = (-8/3) + 2 + 4 - (2/3) y = (1/3) + (1/2) - 2 - (2/3)

[multiply]

y = 2 2/3 y = -1 5/6 combine like terms

result: (-2, 2 2/3) and (1, -1 5/6) are critical points

y" = 2x + 1 take the 2nd derivative

2x + 1 = 0 set the 2nd derivative equal to 0

-1 -1 subtract 1 from each side
____________
2x = -1 subtract
___ ___
2 2 divide each side by 2

x = -1/2 cancel

y(-1/2) = (1/3)(-1/2)³ + (1/2)(-1/2)² - 2(-1/2) - (2/3)

[replace x with -1/2]

y(-1/2) = (1/3)(-1/8) + (1/8) + 1 - (2/3) multiply

y(-1/2) = (-1/24) + (1/8) + 1 - (2/3) multiply

y(-1/2) = (-1/24) + (3/24) + (24/24) - (16/24)

[use equivalent fractions]

y(-1/2) = 10/24 combine like terms

y(-1/2) = 5/12 reduce

result: (-1/2, 5/12) is the point of inflection


(7.) y = sin x cos x here is the problem

y = (1/2)sin 2x double angle id for sine

y' = cos 2x use the chain rule

cos 2x = 0 set the derivative equal to 0

2x = pi/2 2x = 3pi/2 2x = 5pi/2 2x = 7pi/2

[use the unit circle]

x = pi/4 x = 3pi/4 x = 5pi/4 x = 7pi/4

[multiply each side by 1/2, cancel]

y = sin (pi/4) cos (pi/4) replace x with pi/4
_ _
y = (1/√2)(1/√2) use the unit circle

y = 1/2 multiply

y = sin (3pi/4) cos (3pi/4) replace x with 3pi/4
_ _
y = (1/√2)(-1/√2) use the unit circle

y = -1/2 multiply

y = sin (5pi/4) cos (5pi/4) replace x with 5pi/4
_ _
y = -(1/√2)(-1/√2) use the unit circle

y = 1/2 multiply

y = sin (7pi/4) cos (7pi/4) replace x with 7pi/4
_ _
y = (-1/√2)(1/√2) use the unit circle

y = -1/2 multiply

results: (pi/4, 1/2) (3pi/4, -1/2) (5pi/4, 1/2)

and (7pi/4, -1/2) are the extreme points

y" = -2 sin 2x take the 2nd derivative

-2 sin 2x = 0 set the 2nd derivative equal to 0
__________ __
-2 -2 divide each side by -2

sin 2x = 0 divide and cancel

2x = 0 2x = pi 2x = 2pi 2x = 3pi use the unit circle
__ __ ___ ___ ___ ____ ___ ____
2 2 2 2 2 2 2 2 divide ea side by 2

x = 0 , x = pi/2 , x = pi , x = 3pi/2 cancel

y = sin 0 cos 0 y = sin (pi/2) cos (pi/2)

y = sin pi cos pi y = sin (3pi/2) cos (3pi/2)

[replace x with 0, pi/2, pi and 3pi/2]

y = 0 y = 0 y = 0 y = 0 simplify, use the unit circle

results: (0,0) , (pi/2, 0) , (pi, 0) and (3pi/2, 0)
are the points of inflection.

(8.) y = cos² x here is the problem

y = cos² 0 replace x with 0

y = (1)² use the unit circle

y = 1 square 1

result: The y intercept is (0,1)

cos² x = 0 set the function equal to 0

cos x = 0 take square roots

x = pi/2 x = 3pi/2 use the unit circle

result: The x intercepts are (pi/2, 0) and (3pi/2, 0)

y = cos² x here is the problem

y' = -2 sin x cos x use the chain rule

y' = - sin 2x double angle id for sine

-sin 2x = 0 set the derivative equal to 0

sin 2x = 0 multiply each side by -1, cancel

2x = 0 2x = pi 2x = 2pi 2x = 3pi 2x = 4pi

[use the unit circle]

x = 0 x = pi/2 x = pi x = 3pi/2 x = 2pi

[divide each side by 2, cancel]

y = cos² 0 y = cos² (pi/2) y = cos² (pi) y = cos² (3pi/2)

y = cos² 2pi replace x with the critical numbers

y = 1 y = 0 y = 1 y = 0 y = 1 simplify

results: (0,1) , (pi/2, 0) , (pi, 1) (3pi/2, 0) (2pi, 1)

[these are the extreme points]

y' = - sin 2x

y" = -2 cos 2x use the chain rule

-2 cos 2x = 0 set the 2nd derivative equal to 0
___________ ___
-2 -2 divide each side by -2

cos 2x = 0 divide and cancel

2x = pi/2 2x = 3pi/2 2x = 5pi/2 2x = 7pi/2

[use the unit circle]

x = pi/4 x = 3pi/4 x = 5pi/4 x = 7pi/4

[multiply each side by 1/2, cancel]

y = cos² x

y = cos² (pi/4)

y = cos² (3pi/4)

y = cos² (5pi/4)

y = cos²(7pi/4) replace x with these

y = 1/2 y = 1/2 y = 1/2 y = 1/2 simplify

results:

(pi/4, 1/2) (3pi/4, 1/2) (5pi/4, 1/2) (7pi/4, 1/2)

[these are the points of inflection]


(9.) y = (x + 2)²(x - 2)² here is the problem

y = [(x + 2)(x - 2)][(x + 2)(x - 2)] rearrange like this

y = (x² - 4)(x² - 4) foil multiply combine like terms

y = x⁴ - 8x² + 16 foil multiply combine like terms

y' = 4x³ - 16x take the derivative

4x³ - 16x = 0 set the derivative equal to 0
___ ___ __
4 4 4 divide thru by 4

x³ - 4x = 0 divide and cancel

x(x² - 4) = 0 factor

x(x - 2)(x + 2) = 0 factor

x = 0 x - 2 = 0 x + 2 = 0 set each factor equal to 0

+ 2 +2 -2 -2 add this to each side
_______ ____________ ___________
x = 0 , x = 2 , x = -2 add

result: (0,16), (2,0) and (-2,0) are the critical points

y" = 12x² - 16 take the 2nd derivative

12x² - 16 = 0 set the 2nd derivative equal to 0
___ ___ ___
4 4 4 divide thru by 4

3x² - 4 = 0 divide

+ 4 + 4 add 4 to each side
________________
3x² = 4 add
___ ___
3 3 divide each side by 3

x² = 4/3 cancel
_ _
x = 2/√3 x = -2/√3 take square roots

y = x⁴ - 8x² + 16 use this equation

y = (4/3)² - 8(4/3) + 16 replace x² with 4/3

y = (16/9) - (32/3) + 16 multiply

y = (16/9) - (96/9) + (144/9) use equivalent fractions

y = (160/9) - (96/9) add

y = (64/9) combine like terms
_ _
result: (2/√3, 64/9) and (-2/√3, 64/9)

are the points of inflection.

y" = 12x² - 16 use this equation

y"(0) = 12(0)² - 16 y"(-2) = 12(-2)² - 16

y"(2) = 12(2)² - 16 replace x with 0, 2 and -2

y"(0) = -16 y"(-2) = 32 y"(2) = 32 simplify

results: (0,16) is a maximum

(-2, 0) and (2,0) are minimums


(10.) y = 2x(x + 4)³

y = 2x(x³ + 12x² + 48x + 64) cube the binomial

y = 2x⁴ + 24x³ + 96x² + 128x multiply thru parentheses

y' = 8x³ + 72x² + 192x + 128 take the derivative

8x³ + 72x² + 192x + 128 = 0 set the derivative equal to 0
___ ___ ____ ___ __
8 8 8 8 8 divide thru by 8

x³ + 9x² + 24x + 16 = 0 divide and cancel

-1 | 1 9 24 16 use synthetic division
____| -1 - 8 - 16
_______________________
1 8 16 0

x² + 8x + 16 = 0 here is the remaining equation

(x + 4)² = 0 factor

x + 4 = 0 set this factor equal to 0

-4 -4 subtract 4 from each side
______________
x = -4 subtract

y = 2x(x + 4)³ use this equation

y = 2(-1)(-1 + 4)³ y = 2(-4)(-4 + 4)³

[replace x with -1 and with -4]

y = -54 y = 0 simplify

results: (-1, -54) and (-4,0) are the critical points

y' = 8x³ + 72x² + 192x + 128 use this equation

y" = 24x² + 144x + 192 take the 2nd derivative

24x² + 144x + 192 = 0 set the 2nd derivative equal to 0
____ _____ ____ ___
24 24 24 24 divide thru by 24

x² + 6x + 8 = 0 divide

(x + 4)(x + 2) = 0 factor

x + 4 = 0 x + 2 = 0 set each factor equal to 0

-4 -4 - 2 -2 subtract this from each side
__________ ___________
x = -4 , x = -2 subtract

y(-4) = 0 y(-2) = 2(-2)((-2) + 4)³ replace x with -2

y(-2) = (-4)(2)³ multiply add

y(-2) = (-4)(8) cube

y(-2) = -32 multiply

results: (-4,0) and (-2,-32) are points of inflection

y" = 24x² + 144x + 192 use this equation

y"(-1) = 24(-1)² + 144(-1) + 192 replace x with -1

y"(-1) = 24 - 144 + 192 multiply

y"(-1) = 216 - 144 add

y"(-1) = 72 subtract

result: (-1, -54) is a minimum