Question

How do you find the y coordinate of the inflection point of the function `f(x)= 10(x-5)^3+2`?

Answer 1

The inflection points of a function are the zeroes of the second derivative in which the sign of it change in sign.

So let's find the second derivative:

`y'=30(x-5)^2`

and

`y''=60(x-5)`.

Ther is only one zero: `x=5` and the second derivative change in sign in it, so `P(5,2)` is the only inflection point, as you can see from the graph:

graph{10(x-5)^3 +2 [-10, 10, -5, 5]}