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