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

1 Answer
May 19, 2018

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