;BY FEBRUARY 27 YOUR ROOT FINDING MUST HANDLE THE FOLLOWING DIFFERENTIABLE FUNCTIONS....
;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
;CONTINUOUS, no roots 
PARS: f(x) = 2                RESULT: valid                    TREE: C                        
EVAL: f(1.00)  = 2.00000      RESULT: valid                    TREE: C
EVAL: f'(1.00) = 0.00000      RESULT: valid                    TREE: C
EVAL: f''(1.00)= 0.00000      RESULT: valid                    TREE: C
Summary of Roots:
  f:[]
 f':[]
f'':[]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
;CONTINUOUS, no roots
PARS: f(x) = sin(3)           RESULT: valid                    TREE: C                        
EVAL: f(12.00)  = 0.14112     RESULT: valid                    TREE: C
EVAL: f'(12.00) = 0.00000     RESULT: valid                    TREE: C
EVAL: f''(12.00)= 0.00000     RESULT: valid                    TREE: C
Summary of Roots:
  f:[]
 f':[]
f'':[]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
;CONTINUOUS, 1 root: {0.0} 
PARS: f(x) = x                RESULT: valid                    TREE: V                        
EVAL: f(3.00)  = 3.00000      RESULT: valid                    TREE: V
EVAL: f'(3.00) = 1.00000      RESULT: valid                    TREE: C
EVAL: f''(3.00)= 0.00000      RESULT: valid                    TREE: C
Summary of Roots:
  f:[0.0]
 f':[]
f'':[]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
;CONTINUOUS, 1 root: {0.75} 
PARS: f(x) = 4*x-3            RESULT: valid                    TREE: COVOC                    
EVAL: f(12.00)  = 45.00000    RESULT: valid                    TREE: COVOC
EVAL: f'(12.00) = 4.00000     RESULT: valid                    TREE: C
EVAL: f''(12.00)= 0.00000     RESULT: valid                    TREE: C
Summary of Roots:
  f:[0.75]
 f':[]
f'':[]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
;CONTINUOUS, 2 (equal) roots: {0.0} 
PARS: f(x) = x^2              RESULT: valid                    TREE: VOC                      
EVAL: f(-1.00)  = 1.00000     RESULT: valid                    TREE: VOC
EVAL: f'(-1.00) = -2.00000    RESULT: valid                    TREE: COV
EVAL: f''(-1.00)= 2.00000     RESULT: valid                    TREE: C
Summary of Roots:
  f:[0.0]
 f':[0.0]
f'':[]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
;CONTINUOUS, 4 roots: {-1.0, 0.0, 1.0}
PARS: f(x) = x^4-x^2          RESULT: valid                    TREE: VOCOVOC                  
EVAL: f(0.00)  = 0.00000      RESULT: valid                    TREE: VOCOVOC
EVAL: f'(0.00) = 0.00000      RESULT: valid                    TREE: COVOCOCOV
EVAL: f''(0.00)= -2.00000     RESULT: valid                    TREE: COCOVOCOC
Summary of Roots:
  f:[-1.0, 0.0, 1.0]
 f':[-0.71, 0.0, 0.71]
f'':[-0.41, 0.41]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
;CONTINUOUS, 6 (equal) roots: {0.0}
PARS: f(x) = x^6              RESULT: valid                    TREE: VOC                      
EVAL: f(0.00)  = 0.00000      RESULT: valid                    TREE: VOC
EVAL: f'(0.00) = 0.00000      RESULT: valid                    TREE: COVOC
EVAL: f''(0.00)= 0.00000      RESULT: valid                    TREE: COCOVOC
Summary of Roots:
  f:[0.0]
 f':[0.0]
f'':[0.0]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
;CONTINUOUS, 5 roots: {-6.28, -3.14, 0.0, 3.14, 6.28}
PARS: f(x) = sin(x)           RESULT: valid                    TREE: VF                       
EVAL: f(0.00)  = 0.00000      RESULT: valid                    TREE: VF
EVAL: f'(0.00) = 1.00000      RESULT: valid                    TREE: VF
EVAL: f''(0.00)= 0.00000      RESULT: valid                    TREE: VF
Summary of Roots:
  f:[-6.28, -3.14, 0.0, 3.14, 6.28]
 f':[-4.71, -1.57, 1.57, 4.71]
f'':[-6.28, -3.14, 0.0, 3.14, 6.28]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
;CONTINUOUS, 3 roots: {-6.28, 0.0, 6.28}
PARS: f(x) = sin(x/2)         RESULT: valid                    TREE: VOCF                     
EVAL: f(0.00)  = 0.00000      RESULT: valid                    TREE: VOCF
EVAL: f'(0.00) = 0.50000      RESULT: valid                    TREE: VOCFOCOVOCOC
EVAL: f''(0.00)= 0.00000      RESULT: valid                    TREE: VOCFOCOVOCOCOCOVOCOCOVOCFOCOVOCOCOCOVOCOCOC
Summary of Roots:
  f:[-6.28, 0.0, 6.28]
 f':[-3.14, 3.14]
f'':[-6.28, 0.0, 6.28]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
;CONTINUOUS, 9 roots: {-5.5, -3.93, -2.36, -0.79, 0.79, 2.36, 3.93, 5.5, 7.07}
PARS: f(x) = cos(2*x)         RESULT: valid                    TREE: COVF                     
EVAL: f(0.00)  = 1.00000      RESULT: valid                    TREE: COVF
EVAL: f'(0.00) = 0.00000      RESULT: valid                    TREE: COVFOC
EVAL: f''(0.00)= 4.00000      RESULT: valid                    TREE: COVFOCOCOCOVFOC
Summary of Roots:
  f:[-5.5, -3.93, -2.36, -0.79, 0.79, 2.36, 3.93, 5.5, 7.07]
 f':[-6.28, -4.71, -3.14, -1.57, 0.0, 1.57, 3.14, 4.71, 6.28]
f'':[-5.5, -3.93, -2.36, -0.79, 0.79, 2.36, 3.93, 5.5, 7.07]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~