Is f(x)=4x^5-2x^4-9x^3-2x^2-6x concave or convex at x=-1?

1 Answer
Jan 22, 2016

Concave (also called "concave down").

Explanation:

Concavity and convexity are determined by the sign of the second derivative:

  • If f''(-1)<0, then the function is concave at x=-1.
  • If f''(-1)>0, then the function is convex at x=-1.

Find the second derivative:

f(x)=4x^5-2x^4-9x^3-2x^2-6x
f'(x)=20x^4-8x^3-27x^2-4x-6
f''(x)=80x^3-24x^2-54x-4

Find the sign of the second derivative when x=-1:

f''(-1)=80(-1)^3-24(-1)^2-54(-1)-4

=80(-1)-24(1)+54-4=-80-24+50=-54

Since f''(-1)<0, the function is concave at x=-1. This means that it will resemble the nn shape. We can check a graph of f(x):

graph{4x^5-2x^4-9x^3-2x^2-6x [-5, 5, -26.45, 19.8]}