How do you determine the values of \[x\] for which the graph of \[f\] is concave up and those on which it is concave down for \[f\left( x \right) = 6{x^3} - 108{x^2} + 13x - 26\]?

Hint: Concavity of a function is the rate of change of the slope of the function. As we know, the first derivative of a function is the slope of the function. The second derivative of the function defines the concavity of a function. If the second derivative of the function is positive then the graph of the function will be concave up and if the second derivative of the function is negative then the graph of the function will be concave down.

Complete step by step answer:
As we have to determine the values of \[x\] for which the graph of \[f\] is concave up and those on which it is concave down for \[f\left( x \right) = 6{x^3} - 108{x^2} + 13x - 26\].
For this, we need to understand the concept of concavity. Concavity of a function is the rate of change of slope of that function.
Now, given \[f\left( x \right) = 6{x^3} - 108{x^2} + 13x - 26\].
First, we will calculate the first derivative of the function i.e., the slope. So, we get,
\[ \Rightarrow f'\left( x \right) = 18{x^2} - 216x + 13\]
Now, to calculate the concavity of the given function, we need to calculate its second derivative. So, we get,
\[ \Rightarrow f''\left( x \right) = 36x - 216\]
As we know, if the concavity of a function is positive, it depicts that slope of the function is increasing and the graph of the function will be concave up.
So, the graph of \[f\] is concave up for \[f''\left( x \right) > 0\]. So, we can write
\[ \Rightarrow 36x - 216 > 0\]
On simplifying, we get
\[ \Rightarrow 36x > 216\]
As we know that inequality remains the same on dividing both the sides by a positive constant.
On dividing both the sides by \[36\], we get
\[ \Rightarrow x > \dfrac{{216}}{{36}}\]
On simplification, we get
\[ \Rightarrow x > 6\]
Now, if the concavity of a function is negative, it depicts that the slope is decreasing and the graph of the function is concave down.
So, the graph of \[f\] is concave down for \[f''\left( x \right) < 0\]. So, we can write
\[ \Rightarrow 36x - 216 < 0\]
On simplifying, we get
\[ \Rightarrow 36x < 216\]
As we know that inequality remains the same on dividing both the sides by a positive constant.
On dividing both the sides by \[36\], we get
\[ \Rightarrow x < \dfrac{{216}}{{36}}\]
On simplification, we get
\[ \Rightarrow x < 6\]
Therefore, the graph of \[f\] is concave up for \[x > 6\] and concave down for \[x < 6\] for \[f\left( x \right) = 6{x^3} - 108{x^2} + 13x - 26\].

Note:
If a function changes from concave up to concave down or vice versa around a point i.e., points where concavity changes are called points of inflection. Geometrically, if the graph of a function behaves like a portion of the parabola that opens upward then the function is concave upward. Similarly, a function is concave down on an interval if it looks like a portion of the parabola that opens downward.