Question

Is `f(x) =(x-1)^3/(x^2-2)` concave or convex at `x=8`?

Answer 1

f(x) is convex at x=8

Explanation:

`f(x)=(x-1)^3/(x^2-2)`

Differentiating with respect to x,

`f'(x)= ((x - 1)^2 (x^2 + 2 x - 6))/(x^2 - 2)^2=(x^4 - 9 x^2 + 14 x - 6)/(x^2-2)^2`

This function, the derivative, will tell us the slope/gradient of the function at any point x

If we then differentiate again with respect to x,

`f''(x)=(2 (x - 1) (5 x^2 - 16 x + 14))/(x^2 - 2)^3`

This function, the second derivative, will tell us the curvature of the function at any point (i.e. whether it is concave or convex)

Hence, by checking if `f''(8)` is positive or negative, we can find out if the curvature at x=8 is convex or concave.

`f''(8)=(2 ((8) - 1) (5 (8)^2 - 16 (8) + 14))/(8^2 - 2)^3=721/59582>0`

Hence f(x) is convex at x=8