Question

Is `f(x)=x^4-2x^3-9x-14` concave or convex at `x=-1`?

Answer 1

At `x=-1` the curve is Convex or Concave Up Explanation given below.

Explanation:

`f(x) = x^4-2x^3-9x-14`

Step 1: Find the derivative

`f'(x) = 4x^3-6x^2-9`

Step 2: Differentiate again with respect to x

`f^2(x) =12x^2-12x`

Step 3: Substitute `x=-1` in `f^2(x)` and check for sign

`f^2(-1)=12(-1)^2-12(-1)`
`f^2(-1)=12+12`
`f^2(-1)=24`

If `f^2(x) >0` then the curve is convex.
If `f^2(x) <0`, then the curve is *concave *

We can see at `x=-1` the second derivative is greater than zero, hence, the curve is convex.

For further information, you can refer
Note: Concave up is same as convex and concave down is concave

http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx