1. ## concave...

the question is

given f'(x) = (2x)/(x^2 - 4)^2 and f"(x) = -2(3x^2 + 4)/(x^2 - 4)^2
B) Use calculus to find all intervals on which f(x) is concave up or down and the find the x-coordinate of all inflection points, if they exist.

i've tride f"(x) is 0 when -2(3x^2 + 4) = 0 x= ... ?

f"(x) is undefined when (x^2 - 4)^2 = 0, x=2, -2

what should i do with " -2(3x^2 + 4) = 0 x= ... ? " <-- here?

2. Originally Posted by haebinpark
the question is

given f'(x) = (2x)/(x^2 - 4)^2 and f"(x) = -2(3x^2 + 4)/(x^2 - 4)^2
B) Use calculus to find all intervals on which f(x) is concave up or down and the find the x-coordinate of all inflection points, if they exist.

i've tride f"(x) is 0 when -2(3x^2 + 4) = 0 x= ... ?

f"(x) is undefined when (x^2 - 4)^2 = 0, x=2, -2

what should i do with " -2(3x^2 + 4) = 0 x= ... ? " <-- here?
Concave up and down depends on the sign of the second derivative. Inflection points are when $\displaystyle f''(x)=0$.

And $\displaystyle f''(x)=\frac{-2(3x^2+4)}{(x^2-4)^{\color{red}3}}$

So find where $\displaystyle f''(x)<0$ (down) and where $\displaystyle f''(x)>0$ (up).