Question

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

Answer 1

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`.