Find the inflection point(s), if any, of the function.

Question

G(x)=6x³-9x²+3x-2

Recall that an inflection point is a point on the graph of a continuous function where the tangent line exists and where the concavity changes. To find inflection points for the given function, we follow the method below.

1. 1) Compute g''(x).
2. Determine the numbers in the domain of g for which g''(x) = 0 or g''(x) does not exist.
3. Determine the sign of g''(x) to the left and right of each number c found in Step 2. If there is a change in the sign of g''(x) as we move across x = c, then (c, f(c)) is an inflection point of g.

Answer 1

Hi Ernesto,

G(x) = 6x³ - 9x² + 3x - 2

G'(x) = 18x² - 18x + 3

G''(x) = 36x - 18

To find the inflection point, set G''(x) = 0 and solve for x.

36x - 18 = 0

or

36x = 18

which leads to x = 1/2

Let us test around x = 1/2 to see if the sign of G''(x) changes there.

Consider x = 0.495, just a little less than 1/2 = 0.5

G''(0.495) = -0.18

Now, consider x = 0.505, just a little more than 0.5.

G''(0.505) = +0.18

So, the sign of G''(x) changes at x = 1/2, so this is an inflection point.

What is the value of G(x) at x = 0.5?

G(0.5) = -2.

So, the inflection point is (1/2, -2)