What are the points of inflection, if any, of #f(x)=12x^3 -16x^2 +x +7 #?
Find where the second derivative is equal to 0.
Points of inflection in a function are those points where the function changes its concavity, between convex (i.e. concave up), and concave (i.e. concave down). Because a positive second derivative denotes a point where the function is convex and a negative one a point where the function is concave, at a point where concavity changes the second derivative must be 0.
Thus, step 1 is finding the second derivative using the Power Rule:
Now, we set this equal to 0 to find the appropriate x value or values. Since our second derivative is linear, we know there will be only one appropriate x value, and we know that the second derivative will change from positive to negative or negative to positive at that point. Thus, we know it will be an inflection point.
The only inflection point is at
The only point of inflection is at