The given function is `f ( x ) = - x^3 + 6 x^2 - 5 . ` It is defined for all real `x ` and infinitely differentiable for all `x .`
To find its inflection points, consider its first and second derivatives:
`f ' ( x ) = - 3 x^2 +12 x , ` `f '' ( x ) = - 6 x + 12 = -6 ( x - 2 ) .`
A function is concave down where its second derivative is negative and is concave up where its second derivative is positive. For the given function, `f '' ( x ) lt 0 ` for `x in ( 2 , + oo ) ` and `f'' ( x ) gt 0 ` for `x in (- oo , 2 ) .`
An inflection point, by definition, is a point where the concavity changes its direction. A necessary condition is that the second derivative is zero at this point. There is only one such point, `x_0 = 2 , ` and indeed the function is concave up to the left of it and is concave down to the right.
This way, the given function is concave down on `( 2 , + oo ) , ` is concave up on `( - oo , 2 ) , ` and has one inflection point, `x = 2 .`