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Sketch the graph of a twice-differentiable function `y=f(x)` whose derivatives have the following sign patterns below. Also you are given:`f(1)=f(4)=0, \text{ }f'(2)=f'(4)=0,\text{ }f''(3)=0`.

Interval `(-oo,2)` `(2,4)` `(4,oo)`
`f'(x)` `\text{ }-` `\text{ }+` `\text{ }-`

Interval `(-oo,3)` `(3,oo)`
`f''(x)` `\text{ }+` `\text{ }-`


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X
MathQuill
   x  x  √  n√  |   | (   ) π ∞ DNE
   x  √  (   ) π ∞ DNE
( ) ( ] [ ) [ ] —∞ ∞ ∪ DNE
< > ≤ ≥ or All Real Numbers DNE
log logn ln n√  | | e 
sin cos tan arcsin arccos arctan ⟨ ⟩
[more..]