Question

Is `f(x)=-8x^5-x^2+5x-4` concave or convex at `x=-4`?

Answer 1

convex at x = -4

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) = -8x^5 - x^2 + 5x - 4`

`f'(x) = -40x^4 - 2x + 5`

and `f''(x) = -160x^3 - 2`

`rArr f''(-4) = -160(-4)^3 - 2(-4) = 10240 + 8 = 10248`

since f''(-4) > 0 then f(x) is convex at x = -4
graph{-8x^5-x^2+5x-4 [-18.02, 18.03, -9.01, 9.01]}