Question

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

Answer 1

concave at x = -1

Explanation:

To test if a function is concave / convex at f(a) , require to find the value of f''(a).

• If f''(a) > 0 then f(x) is convex at x = a

• If f''(a) < 0 then f(x) is concave at x = a

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

f'(x) `= 3x^2 - 4x -x^-2`

and f''(x)`= 6x - 4 +2x^-3 = 6x-4 + 2/x^3`

so f''(-1) `= 6(-1) - 4 + 2/(-1)^3 = -6-4-2 = -12`

Since f''(-1) < 0 then f(x) is concave at x = -1
graph{x^3-2x^2+1/x [-12.66, 12.65, -6.33, 6.32]}