Question

Is `f(x)=3x^5-x^3+8x^2-x+3` concave or convex at `x=-4`?

Answer 1

Concave down

Explanation:

To find the concavity of a function `f(x)`, you will need to determine the second derivative `f''(x)`, and then evaluate at at the given point. If the result of `f''(a) > 0`, the function is concave up; if the result is negative, it it concave down.

`f(x) = 3x^5 - x^3 + 8x^2 - x + 3`

`f'(x) = 15x^4 - 3x^2 + 16x = 1`

`f''(x) = 60x^3 - 6x + 16`

`f''(-4) = 60(-4)^3 - 6(-4) + 16 = -3,800 < 0`

Thus, `f(x)` is concave down at `x = -4`

graph{3x^5 - x^3 + 8x^2 - x + 3 [-4.59, 2.72, -5000.82, 500.84]}