CPhill

x / r =  [ x + h ] / R

x (R / r ) = x + h

x (R/r) - x =  h

x [( R/r) - 1 ]   = h

x  =    h / [ (R/r) - 1 ]

H = x + h

H = h / [ (R/r) -1 ] + h

"Anchor" any one of the groups  and  arrange them in two ways

Choose any one of the  three remainig groups and arrange this  group in two ways

Choose any one of the two remaing groups and arrange this  group in two ways

Arrange the remaining group in two ways

2 * C(3,1) * 2 * C(2,1)* 2 * 2  =   96 ways

7 times  =  10

6 times  =  10 * 9 * 7  =  630

10 + 630  =  640 

3210

510

420

321

60

51

42

AB / BD    =  AC / DC

6 / 3  = AC /  5

2  =   AC / 5

AC = 10

  x^2  / ( x + 2)  =   x^2 (x + 2)^(-1)  = f(x)

Take the derivative and set to 0

f'(x)  =  2x (x + 2)^(-1) - x^2 (x + 2)^(-2)  = 0

(x + 2)^(-2) [ 2x ( x + 2) - x^2]  =  0

x^2 + 4x  =  0

x ( x + 4)  = 0

Local Max occurs at x = -4

Local Min occurs at x = 0

No min for  x >  2

$f(x) = \frac{x^4 + 2x^3 + 3x^2 + 2x + 1}{x}.$

Simplify as

f(x)  =  x^3 + 2x^2 + 3x + 2 + x^(-1)        take the derivative and set to 0

3x^2 + 4x + 3 - x^(-2)  =  0      multiply through by x^2

3x^4 + 4x^3 + 3x^2 - 1  =  0

Graphing this shows a local min  at   x ≈  .43426

$\frac{1}{x} \ge 3 - 2x$

1  ≥ 3x - 2x^2

2x^2 -3x + 1  ≥  0       set as an equality  to find test intervals

2x^2 - 3x + 1   =  0

x =   [ 3 + sqrt [ 9 - 8]] / 4  =  1     or    x =  [ 3 - sqrt [ 9 -8 ] ] / 4] =  1/2

Solution

0 < x  ≤ 1/2   ∪    x  ≥ 1

$f(x) = \log (-20x + 16 \sqrt{x} - x)$

f'x =  1  / [ -20x + 16sqrt x - x] * ln 10  =   ln 10 * ( -20x + 16sqrt x - x)^-1 * [ -20 + 8 / x^(1/2) - 1 ]

Simplify and set to 0 to  0       

-20  + 8 / (x)^(1/2) - 1  =  0 

-21 + 8 / x^(1/2)  = 0

8/ x^(1/2)   = 21

x^(1/2)  = 8/21          square both sides

x =  (64 / 441)

2 + sqrt 2   +  1/(2 + sqrt 3) * ( 2-sqrt 3)/(2 -sqrt 3)  +  1/ (sqrt 5 - 2) * (sqrt 5 + 2)/(sqrt 5 +2)  =

2 + sqrt 2  + 2 - sqrt 3  +  sqrt 5 + 2   =

6 + sqrt 2 - sqrt 3 + sqrt 5

2^(16x) = 128 * 4^x

2^(16x)  2^7 * (2^2)^x

2^(16x)  = 2^7 * 2^(2x)

2^(16x)  = 2^(7 + 2x)

16x  = 7 + 2x

14x  = 7

x = 7/14 =  1/2

2^x  = 2^(1/2)  =  sqrt 2

a + b  = 14

b = 14  - a

a^3  + (14 -a)^3  = 812 + a^2 + (14 -a)^2

a^3  - a^3 + 42a^2 -588a + 2744  = 812 + a^2 + a^2 - 28a + 196

40a^2  - 560a + 1736  =  0 

5a^2  -  70a + 217  =  0

a  =  [  70  +  sqrt  [  70^2  - 20*217] ] / 10   =  [ 70 + sqrt [ 15260] ] / 10  =  [ 70 + 2sqrt 35 ] / 10  =

7 +  sqrt (35) / 5

And  b =  7 - sqrt(35) / 5

Also   a =  7 - sqrt (35) / 5     and b = 7 + sqrt (35)   / 5