Question

For what values of x is the graph of y equals negative 3 divided by the quantity 4 plus x concave downward?

Answer 1

We are given our function as

$y=\frac{-3}{x+4}$

For finding concavity , firstly we will find second derivative

$y'=\frac{d}{dx}\left(\frac{-3}{x+4}\right)$

$=-3\frac{d}{dx}\left(\frac{1}{x+4}\right)$

$=-3\left(-\frac{1}{\left(x+4\right)^2}\right)\cdot \:1$

$y'=\frac{3}{\left(x+4\right)^2}$

now, we can find derivative again

$y''=\frac{d}{dx}\left(\frac{3}{\left(x+4\right)^2}\right)$

$=3\frac{d}{dx}\left(\left(x+4\right)^{-2}\right)$

$y''=3\left(-\frac{2}{\left(x+4\right)^3}\right)\cdot \:1$

$y''=-\frac{6}{\left(x+4\right)^3}$

now, we can know second derivative is undefined when denominator =0

so, we set denominator =0

and then we can solve for x

$x+4=0$

$x=-4$

now, we can draw a number line and locate x=-4

and then we can find sign of second derivative on each intervals

so,

Concave downward interval:

$(-4,\infty)$