Is f(x)=-5x^5-2x^4-2x^3+14x-17 concave or convex at x=0?
1 Answer
Neither. It is a point of inflection.
Explanation:
Convexity and concavity are determined by the sign of the second derivative.
- If
f''(0)>0 , thenf(x) is convex whenx=0 . - If
f''(0)<0 , thenf(x) is concave whenx=0 .
Find the function's second derivative.
f(x)=-5x^5-2x^4-2x^3+14x-17
f'(x)=-25x^4-8x^3-6x^2+14
f''(x)=-100x^3-24x^2-12x
Find the sign of the second derivative at
f''(0)=0
Notice that the sign of the second derivative is neither positive nor negative. This means that the function is neither convex nor concave. This means that is may be a point of inflection.
We can check a graph of the function:
graph{-5x^5-2x^4-2x^3+14x-17 [-2.5, 2.5, -120, 100]}
Graphically,