Find the open intervals on which f is concave up (down). Then determine the x-coordinates of all inflection points of f.

Question

Let f(x)=(7x-2)/(x+2). Find the open intervals on which f is concave up (down). Then determine the x-coordinates of all inflection points of f. 

1. f is concave up on which intervals?

2. f is concave down on which intervals?

3. the inflection points occur at x =?

 

Please provide final answers. 

Answer 1

To get the concavity, we need to take the second derivative of the function.

 

First, take f'(x) = 16/(x+2)^2 using the quotient rule

Next, take f''(x) = -32/(x+2)^3 using the quotient rule on the first derivative

 

Solving f''(x) = 0 would normally give the point of inflection, but in this case we'll get no solution, so we will inspect the critical value of x = -2.

 

Next, check a value smaller than x = -2 and a value larger than x = -2 to test for concavity (negative implies concave down).  

 

Let me know if this gets you going in the right direction!