Answer:

The vertex is 
Step-by-step explanation:
For a general quadratic function the form is:

For the function

Take common factor 3.

The values of the coefficients for the function within the parenthesis are the following:
,
, 
Take the value of b and divide it by 2. Then, the result obtained squares it.


Add and subtract 
![f(x) = 3([x ^ 2 -\frac{4}{3}x +\frac{4}{9}]- 2-\frac{4}{9})](https://tex.z-dn.net/?f=f%28x%29%20%3D%203%28%5Bx%20%5E%202%20-%5Cfrac%7B4%7D%7B3%7Dx%20%2B%5Cfrac%7B4%7D%7B9%7D%5D-%202-%5Cfrac%7B4%7D%7B9%7D%29)
Write the expression of the form


The vertex is 
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Differentiate the given solution:

Substitute
and
into the ODE:

and it's easy to see the left side indeed reduces to 0.
It the property of distributive