Question

f(x)=(5x-6)/(x+4) so the second derivative is (x^2-44x-192)/(x+4)^4 so x= 48 and -4. i found that f is concave up on the interval (-inf, -4), but i can't figure out when it is concave down and the inflection points

Answer 1

if it is not concave up, it is concave down

Answer 2

your second derivative is wrong however

Answer 3

\[f(x)=\frac{5x-6}{x+4}\]
\[f'(x)=\frac{26}{(x+4)^2}\]
\[f''(x)=-\frac{52}{(x+4)^3}\]

Answer 4

and so \(f''>0\iff x<-4\) which is where it is concave up, then concave down on \((-4,\infty)\)

Answer 5

ohhh okay thank you!!!!!!!

Answer 6

and so there are no inflection points?

Answer 7

don't forget, you know something about rational functions without ever looking at calculus. you have a vertical asymptote at \(x=-4\) and a horizontal one at \(y=5\) so your function looks something like this

Answer 8

actually here is a better picture
http://www.wolframalpha.com/input/?i=%285x-6%29%2F%28x%2B4%29
you see where it is concave up and concave down. it changes concavity at the vertical asymptote, so there is no inflection point

Answer 9

ohh okay that makes sense