Let f(x)=-x^4-2x^3+6x-4. Find the intervals on which f is concave up/down. And find the x-coordinates of all inflection points of f

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Given `f(x)=x^4-2x^3+6x-4` we are asked to find the concavity intervals and inflection points:

If the second derivative of a function is positive the function is concave up; if the second derivative is negative the function is concave down.

Inflection points occur when the sign of the second derivative changes.

So we find the second derivative:

`f'(x)=4x^3-6x^2+6`

`f''(x)=12x^2-12x`

Set the second derivative equal to zero:

`12x^2-12x=0==>12x(x-1)=0==>x=0,x=1`

Consider the sign of the second derivative on the following intervals:

`f''(x)>0 (-oo,0)`

`f''(x)<0 (0,1)`

`f''(x)>0 (1,oo)`

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**The function is concave up on `(-oo,0),(1,oo)` and concave down on (0,1). The points of inflection are (0,-4) and (1,1)**

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The graph:

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