Calculus: Early Transcendentals, Chapter 4, 4.3, Section 4.3, Problem 59

When using a graph of f"(x), we follow concavity test:
A a section of function f(x) is concave up when f"(x) > 0.
This means the segment of the graph of f"(x) is above the x-axis.
A a section of function f(x) is concave down when f"(x) < 0.
This means the segment of the graph of f"(x) is below the x-axis.
Using computer algebra system, here is the graph of f(x) = (x^4+x^3+1)/(sqrt(x^2+x+1))



For its second derivative graph:
f"(x) =

The graph of f"(x) intersects the x-axis at x~~ -0.7 and x~~ 0.1.
These will be used as the boundary values to set the intervals of concavity.
These are the final answers.
Concave up: (-oo ,-0.7) and (0.1, +oo )
Concave down: (-0.7, 0.1)

In case you prefer a more accurate data, here is maximize view of the graph of f"(x).


The graph of f"(x) intersects the x-axis at x=-0.64 and x=0.03.
These will be used as the boundary values to set the intervals of concavity.
Concave up: (-oo ,-0.64) and (0.03, +oo )
Concave down: (-0.64, 0.03)