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 atx=-1 . - If
f''(-1)>0 , then the function is convex atx=-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
f''(-1)=80(-1)^3-24(-1)^2-54(-1)-4
=80(-1)-24(1)+54-4=-80-24+50=-54
Since
graph{4x^5-2x^4-9x^3-2x^2-6x [-5, 5, -26.45, 19.8]}