Is f(x)=-5x^5-2x^4-2x^3+14x-17 concave or convex at x=0?

1 Answer
Jan 17, 2016

Neither. It is a point of inflection.

Explanation:

Convexity and concavity are determined by the sign of the second derivative.

  • If f''(0)>0, then f(x) is convex when x=0.
  • If f''(0)<0, then f(x) is concave when x=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 x=0.

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, x=0 does appear to be a point of inflection (the concavity shifts). This is testable by seeing if the sign of the second derivative goes from positive to negative or vice versa around the point x=0.