Question

Determine the open intervals on which the graph is concave upward and concave downward. f(x)= (x^2)/((x^2)+9)

Answer 1

Do you know derivatives are related to finding the concavity of a function over an interval?

Answer 2

yes I found the first derivative to be -2^3/((x^2)+9)^2.... and the second derivative to be -54x^2/((x^2)+9)^2 I just don't understand the next steps

Answer 3

You need to apply Quotient Rule for the first derivative.
Something is missing in your first derivative.

Then you get the second derivative and do the sign chart for it.

Answer 4

for the first derivative i first rewrote the function as \[(x ^{2})(x ^{2}+9)^{-1}\] then the derivative \[(-x ^{2})(x ^{2}+9)^{-2}(2x)\] simplifying as \[-2x ^{3}/(x ^{2}+9)^{2}\]