ElectricPavlov

Here is the funtion and the graph of the derivative  and the line  x = ln4/(.2)

I think ROM will get it.....I used an online derivative calculator to find the result.....he can probably do it in his head ! 

     2^(1/3)

-3 * 2^(1/3)     * (x-1)  + 7       or  maybe       -3 * 2^(1/3)  * ((x-1)+7 )

Sum of a number and -5    ====   (x+-5)

    is multiplied by  -7        ======  7(x-5)

       the result is 49      =========== -7(x-5) = 49    use distributive property

                                                            -7x + 35 = 49    subtract 35 from both sides of the equation

                                                             -  7x = 14        divide both sides by -7

                                                                 x = -2 

I believe it is the 3rd one.....I looked at it graphically, but I do not know exactly how to find the answer otherwise.....I'll think about it for a while and if I can come up with the REASON for the answer , I will re-post....    sorry   ...

If you know how to do the derivative, it is where the slope goes from positive to negative.....

(or as ROM points out below....IF you know how to do the second derivative, it is where this = 0  )

Straight line distance will be

sqrt [   (x1-x2)^2   +   (y1-y2)^2  ]    =   sqrt ( -6-4)^2 + (3-3)^2 )  = sqrt100 = 10 units

Put in -3 for x     

8/ ( 1 + 3e^-(.7(-3)) = .31374...

Area of a triangle = 1/2 base x height    b = b    h = b-5    area = 42 in^2  (given)

42 = 1/2   b (b-5)

84 = b^2-5b

b^2-5b - 84 = 0    factor    (or use quadratic formula)

(b+7)(b-12)= 0      so b  can =  12 or -7      toss out -7 because you can't have a negative length

so b = 12

h = b-5 = 12-5 = 7   

Check:  Area  = 1/2 b h = 1/2  12  7  = 42    Correct !

Here is how I would do the second one:

find slope  =  rise/run  = y1-y2 / x1-x2  =  (5- -7)/(2-6) = 12/-4 = -3

Perpindicular slope is    -  1/(-3) = 1/3

Find the midpoint of AB

  x      (2+6)/2 = 4

  y      (5+ -7)/2 = -1      so midpoint is    (4,-1)   and slope is 1/3

            line is     y = 1/3  x  + b     sub in your midpoint to calculate  'b'

                          -1 = 1/3 (4) + b    results in b = -7/3

y = 1/3 x -  7/3

Here is a picture: