Determine whether the function \(\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}}}{{{x}^{{{2}}}+{2}}}}\) is concave

Reuben Brennan

Reuben Brennan

Answered question


Determine whether the function f(x)=xx2+2 is concave up or concave down and its intervals?

Answer & Explanation



Beginner2022-03-27Added 17 answers

Step 1
We have that f(x)=xx2+2 calculating its second derivative we find that
So we need to see how the signs change of 2x(x26) as x goes from to +
So from (, 6] we have that f(x)<0
from [6, 0] we have that f(x)>0
from [0, 6] we have that f(x)<0
from [6, +) we have that f(x)>0
In order to determine concavity we use the following theorem
Concavity Theorem:
If the function f is twice differentiable at x=c, then the graph of f is concave upward at (c; f(c)) if f(c)>0 and concave downward if f(c)<0

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