Question

For what values of x is `f(x)= -x^2-x+4` concave or convex?

Answer 1

x = -1/2 with downward concavity

Explanation:

Concavity is expressed by the second derivative.
To start, we must take the first derivate to find the POINT(S):

`f^'(x) = -2x - 1 + 0`

What we just found are the extrema which we can simplify to mean:
set y = 0
0 = -2x - 1
2x = -1
x = -1/2

Then take the second derivate by taking the derivative of the first derivative (`f^'(x)`) to find the CONCAVITY:

`f^''(x) = -2`
Because the result is negative, we know that the graph faces downward like a frowning face.

To solidify this answer, take a look at the graph. graph{-x^2 - x + 4 [-10, 10, -5, 5]}