A function and its graph are given.Use the second derivative to locate all x-values of points of inflection on the graph of Check these results against the graph shown.

Question

​A function and its graph are given.Use the second derivative to locate all x-values of points of inflection on the graph of  Check these results against the graph shown. ​
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Quizplus · Accepted Answer

To find the points of inflection, we need to find the values of x where the concavity changes.
First, we need to find the second derivative of the given function:
f(x) = x^4 - 4x^3 + 6x^2
f'(x) = 4x^3 - 12x^2 + 12x
f''(x) = 12x^2 - 24x + 12 = 12(x^2 - 2x + 1) = 12(x-1)^2
The second derivative is always positive, so the function is concave up on the entire domain.
Therefore, there are no points of inflection on the graph of the function.
Looking at the graph given, we can see that it is consistent with this result, as there are no points of inflection.

​A Function and Its Graph Are Given

A Function and Its Graph Are Given