[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
_______________
____     _____
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
_________________
___        ___
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
_____________
___      ____
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
_____________
____    _____
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
________________
___     ___
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)(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