Question

How do you determine the concavity for `f(x) = x^4 − 32x^2 + 6`?

Answer 1

The concavity of a function is the sign of its second derivative.
If, in a set, it is positive, than the concavity is up, if negative the concavity is down, if it is zero, there could be an inflection point there.

So:

`y'=4x^3-64x`

`y''=12x^2-64`,

than

`12x^2-64>0rArrx^2>64/12rArrx^2>16/3rArr`

`x<-4/sqrt3vvx>4/sqrt3`, or , better:

`x<-4/3sqrt3vvx>4/3sqrt3`. In this set the function has concavity up, in the complementary set it has concavity dawn, in `+-4/3sqrt3` there are two inflection points.