Question

What are the points of inflection, if any, of #f(x)=12x^3 -16x^2 +x +7 #?

Answer 1

Find where the second derivative is equal to 0.

Explanation:

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:

`f(x) = 12x^3 - 16x^2 + x + 7 -> (df)/dx = 36x^2-32x+1 -> (d^2f)/dx^2 = 72x - 32`

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.

`(d^2f)/dx^2 = 72x-32 = 0 -> 72x = 32 -> x = 32/72 = 16/36 = 8/18 = 4/9`

The only inflection point is at `x=4/9`, and the y coordinate is...

`f(4/9) = 12(4/9)^3 -16(4/9)^2 + 4/9 + 7 = 12(64/729)-16(16/81) + 4/9 + 7 ~5.337`

The only point of inflection is at `(4/9, 5.337)`