CPhill

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