# Thread: Interval question..

1. ## Interval question..

I don't get this..

Find all intervals of increasing&decreasing&all relative extrema. Find all intervals where f is concave up or down. Find all points of inflection.

f(x) = 6/(x^2+3)

I am lost on this, no idea how to solve it, I took the derivative and I get my x as sqrt(6i).

How do I get the concept of these problems? Sometimes I'm able to do them and other times I don't get the concept..

2. $\displaystyle f(x) = \frac{6}{x^2+3}$

$\displaystyle f'(x) = \frac{-12x}{(x^2+3)^2}$

$\displaystyle f'(x) = 0$ at $\displaystyle x = 0$

$\displaystyle f'(x) > 0$ for $\displaystyle x < 0$ ... $\displaystyle f(x)$ is increasing on the interval $\displaystyle (-\infty,0)$

$\displaystyle f'(x) < 0$ for $\displaystyle x > 0$ ... $\displaystyle f(x)$ is decreasing on the interval $\displaystyle (0, \infty)$

$\displaystyle f(x)$ has an absolute maximum of $\displaystyle y = 2$ at $\displaystyle x = 0$

$\displaystyle f''(x) = \frac{36(x^2-1)}{(x^2+3)^3}$

$\displaystyle f''(x) = 0$ at $\displaystyle x = 1$ and $\displaystyle x = -1$

$\displaystyle f''(x) > 0$ for $\displaystyle x < -1$ ... $\displaystyle f(x)$ is concave up

$\displaystyle f''(x) < 0$ for $\displaystyle -1 < x < 1$ ... $\displaystyle f(x)$ is concave down

$\displaystyle f''(x) > 0$ for $\displaystyle x > 1$ ... $\displaystyle f(x)$ is concave up

since $\displaystyle f''(x)$ changes sign at $\displaystyle x = 1$ and $\displaystyle x = -1$, there are inflection points at those x-values.