Is f(x)=3x^5-x^3+8x^2-x+3f(x)=3x5x3+8x2x+3 concave or convex at x=-4x=4?

1 Answer
Oct 3, 2017

Concave down

Explanation:

To find the concavity of a function f(x)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]}