Question

Is `f(x)=-2x^5-2x^3+3x^2-x+3` concave or convex at `x=-1`?

Answer 1

Convex.

Explanation:

You can tell if a function is concave or convex by the sign of its second derivative:

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

To find the second derivative, apply the power rule to each term twice.

`f(x)=-2x^5-2x^3+3x^2-x+3`

`f'(x)=-10x^4-6x^2+6x-1`

`f''(x)=-40x^3-12x+6`

Find the sign of the second derivative at `x=-1:`

`f''(-1)=-40(-1)^3-12(-1)+6`

This mostly becomes a test of keeping track of your positives and negatives.

`f''(-1)=-40(-1)+12+6=40+18=58`

Since this is `>0`, the function is convex at `x=-1`. Convexity on a graph is characterized by a `uu` shape.

We can check the graph of the original function:

graph{-2x^5-2x^3+3x^2-x+3 [-2.5, 2, -30, 30]}