I'm having difficulties on this question and I could use some help. Thanks

A metal beam, 10 m long, is unevenly loaded. As a result it is bent slightly from the horizontal. The relationship between the deviation from the horizontal (D mm) and the distance along the beam (x m) is given by:

D= [x(10-x)(x-20)]/1000
At what point along the beam is the deviation from the horizontal maximum?

Expand out D, then it's easy to differentiate.

D` means first derivative, D`` 2nd derivative.

For Maximum, D` = 0 and D`` = negative.

D = [x(10-x)(x-20)]/1000

1000 D = x (-x^2 + 30 x -200)

= -x^3 +30 x^2 - 200x

Differentiating:

1000D` = -3x^2 +60 x- 200

1000D`` = -6x + 60

D` = 0 gives 2 two values of x

For MAXIMUM, Select the value which makes D`` (Second derivative) negative value.

Which value of x do I find out... the second derative one, and from there then what? Thanks

D` = 0

means: -3x^2 +60 x- 200 = 0

Get 2 values of x from this quadratic.

Select one which makes D`` = (-6x + 60)/1000 NEGATIVE.