How do you determine whether the function #f(x)= x/ (x^2+2)# is concave up or concave down and its intervals?

Question

Emma Aldridge · Accepted Answer

Refer to explanation

We have that #f(x)=x/(x^2+2)# calculating its second derivative we find that #d^2 f(x)/(d^2x)=(2x(x^2-6))/(x^2+2)^3#. So we need to see how the signs change of #2x(x^2-6)# as x goes from #-oo# to #+oo# So from #(-oo,-sqrt6]# we have that #f''(x)<0# from #[-sqrt6,0]# we have that #f''(x)>0# from #[0,sqrt6]# we have that #f''(x)<0# from #[sqrt6,+oo)# 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# .

Emily Ammerman · Answer

undefined To determine the concavity of the function $ f(x) = \frac{x}{x^2 + 2} $ and its intervals, we need to find its second derivative and analyze its sign.

First, find the first derivative:
$ f'(x) = \frac{(x^2 + 2) - x(2x)}{(x^2 + 2)^2} $
$ f'(x) = \frac{x^2 + 2 - 2x^2}{(x^2 + 2)^2} $
$ f'(x) = \frac{2 - x^2}{(x^2 + 2)^2} $

Now, find the second derivative:
$ f''(x) = \frac{(x^2 + 2)^2(2x) - (2 - x^2)(2(x^2 + 2)(2x))}{(x^2 + 2)^4} $
$ f''(x) = \frac{2x(x^2 + 2)^2 - (2 - x^2)(4x(x^2 + 2))}{(x^2 + 2)^4} $
$ f''(x) = \frac{2x(x^2 + 2)^2 - 4x(x^2 + 2)(2 - x^2)}{(x^2 + 2)^4} $
$ f''(x) = \frac{2x(x^2 + 2)^2 - 8x(x^2 + 2) + 4x^3(x^2 + 2)}{(x^2 + 2)^4} $
$ f''(x) = \frac{2x(x^4 + 4x^2 + 4) - 8x(x^2 + 2) + 4x^3(x^2 + 2)}{(x^2 + 2)^4} $
$ f''(x) = \frac{2x^5 + 8x^3 + 8x - 8x^3 - 16x + 4x^5 + 8x^3}{(x^2 + 2)^4} $
$ f''(x) = \frac{6x^5 + 8x}{(x^2 + 2)^4} $

Now, we analyze the sign of $ f''(x) $ to determine concavity. If $ f''(x) > 0 $, the function is concave up, and if $ f''(x) < 0 $, it's concave down.

Since $ 6x^5 + 8x $ is always positive for all real numbers $ x $, $ f''(x) > 0 $ for all $ x $. Therefore, the function $ f(x) = \frac{x}{x^2 + 2} $ is concave up for all real numbers. There are no intervals of concavity change.