An

This graph is betwee [-1,1]

`f'(x) = -1/x^2 - 14/x^3 - 3/x^4`

`f''(x) = 2/x^3 + 42/x^4 + 12/x^5`

The function is increasing when f'(x)>0.

f'(x) > 0 when `-1/x^2 - 14x^3 - 3/x^4 gt 0`

Multiply by `-x^4` to get

`x^2 + 14x + 3 lt 0`

We...

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An

This graph is betwee [-1,1]

`f'(x) = -1/x^2 - 14/x^3 - 3/x^4`

`f''(x) = 2/x^3 + 42/x^4 + 12/x^5`

The function is increasing when f'(x)>0.

f'(x) > 0 when `-1/x^2 - 14x^3 - 3/x^4 gt 0`

Multiply by `-x^4` to get

`x^2 + 14x + 3 lt 0`

We get `(x-(-14+-sqrt(14^2 - 4(1)(3)))/2)` .

This happens when `x = (x+7+sqrt(184)/2)` or `(x+7-sqrt(184)/2)`

Simplifying we get `(x+7+sqrt(46))(x+7-sqrt(46))` . This is <0 (and the function is deceasing) on the intervals `(-oo,-7-sqrt(46)), (-7+sqrt(46, 0), (0, +oo)` .

Increasing >0 on the intervals `(-7-sqrt(46),-7+sqrt(46))`

Concave up when f''(0) > 0 and concave down when f''(0) < 0.

`f''(x) = 2/x^3 + 42/x^4 + 12/x^5 gt 0`

`2x^2 + 42x + 12 gt 0`

`x^2 + 21x + 6 gt 0`

`x = (-21+-sqrt(21^2-4(1)(6)))/2 = (-21+-sqrt(417))/2`

`(x-(-21-sqrt(417))/2)(x-(-21+sqrt(417))/2)lt0`

When `(-oo,(-21-sqrt(417))/2)` , `((-21+sqrt(417))/2,0)` the function is concave down,

When `((-21-sqrt(417))/2,(-21+sqrt(417))/2)` and `(0,oo)` it is concave up.