Question

Is `f(x)=-x^5-x^4-2x^3+x-7` concave or convex at `x=-2`?

Answer 1

convex at x = -2

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^5 - x^4 - 2x^3 + x - 7`

f'(x)`= -5x^4 - 4x^3 - 6x^2 + 1`

and f''(x)`= -20x^3 - 12x^2 - 12x`

`rArr f''(-2) = -20(-2)^3 - 12(-2)^2 - 12(-2)`

= 160 - 48 + 24 = 136

since f''(-2) > 0 then f(x) is convex at x = -2
graph{(-x^5-x^4-2x^3+x-7) [-16.02, 16.02, -8.01, 8.01]}