Determine the inflection points of $f(x)=4x^3-\frac{1}{5}x^2-3x+10$.

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Lesson Parent: 
Geometry of Functions XI: Inflection Points
Problem: 

Determine the inflection points of $f(x)=4x^3-\frac{1}{5}x^2-3x+10$.

Answer: 

$x=\frac{1}{60}$ is the inflection point.

SAMPLE SOLUTION

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Solution: 

First, we know that the second derivative is <Sign in to see all the formulas>. Next, let's find when the second derivative is zero:

<Sign in to see all the formulas>

The point <Sign in to see all the formulas> is only an educated guess of the inflection point. We now need to see if it is by proving it. To prove it we must show that its concavity changes around the point <Sign in to see all the formulas>. Recall that when <Sign in to see all the formulas> that the function is concave up and when <Sign in to see all the formulas> it is concave down.

First, let's look at all points such that <Sign in to see all the formulas>. Now let's do some algebra:

<Sign in to see all the formulas>


Therefore, when <Sign in to see all the formulas> the second derivative is guaranteed to be less than zero.

Next, let's look at all points such that <Sign in to see all the formulas>. Now let's do some algebra:

<Sign in to see all the formulas>


So when <Sign in to see all the formulas> the second derivative is guaranteed to be greater than zero.

We have now proven that <Sign in to see all the formulas> is an inflection point by showing that it is concave down (<Sign in to see all the formulas>) to the left and concave up (<Sign in to see all the formulas>) to the right. This is precisely the definition of an inflection point. To be more precise, it is concave down on the interval <Sign in to see all the formulas> and concave up on the interval <Sign in to see all the formulas>.