Question

Is `f(x) =(x-2)^2(x+6)(x-3)` concave or convex at `x=0`?

Answer 1

`f''(0)=-52<0quad=>quad` function is concave `nnn`

Explanation:

`f(x)=(x−2)^2(x+6)(x−3)`

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

`f'(x)=(x−2)(4x^2+5x−42)`

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`f''(x)=(4x^2+5x−42)+(x−2)(8x+5)`

`f''(0)=(−42)+(−2)(+5)=-42-10=-52`

`f''(0)<0quad=>quad` function is concave `nnn` ("spill the water")