How to Find the Inflection Points of this Function

Consider the function: 6x^2 + 5x - 4 / 2x^2 - 7x - 4. Find the point(s) of inflection.


Any help with this problem would be greatly appreciated!

The denominator of this rational function when factored is (2x+1)(x-4).
Thus the function is discontiuous at x = -.5 and at x = 4
In the interval (-infinity, -.5) the function is everywhere concave down so no inflection point in this interval.
In the interval (4, infinity)the function is everywhere concave up so no inflection point in this interval.
Thus only in the interval (-.5. 4) can there be an inflection point.
The inflection point is at approximately x = .5

There are three types of "critical points":
maximum (local or global)
minimum (local or global)
inflection

All three types share one property: the slope of the graph is exactly zero at the critical point.

The slope is found by taking the derivative.

If the point is a maximum, the slope is positive just before, zero at the exact maximum, and negative just after. Thus, the "slope of the slope" is decreasing.
The minimum goes the other way ( - 0 + )
and the inflection point is "wierd", in that the "slope of the slope" is exactly zero

So take the second derivative at the critical point.
if that value is negative, then the critical point is a maximum
if that value is positive, then the critical point is a minimum
if that value is zero, then the critical point is an inflection point.

To make things "easier" for you, you are given a division of two functions, so you will have to use the chain rule for divisions.

step one:
find the derivative, make that derivative equal to zero and find the solutions. All these will be "critical points"; set aside these values.

step 2
find the derivative of the derivative, and evaluate it only at the points you found in step 1.