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------------------------------------------------ Consider the function [tex]f(x) = 12x^5 + 45x^4 - 80x^3 + 6[/tex]. The function [tex]f(x)[/tex] has inflection points at [tex]x = D[/tex], [tex]x = E[/tex], and [tex]x = F[/tex], where [tex]D[/tex], [tex]E[/tex], and [tex]F[/tex] are specific values to be determined.

For each of the following intervals, determine whether the function is concave up or concave down.

To locate the inflection points of the function f(x) = 12x^5 + 45x^4 - 80x^3 + 6, the second derivative f''(x) must be found and set to zero, then solve for x. Each solution is checked for changes in the sign of f''(x) to verify inflection points, which indicate a change in concavity of the function's graph.

Explanation:

To find the inflection points of the function f(x) = 12x5 + 45x4 - 80x3 + 6, we need to examine the places where the second derivative of the function changes sign, which occurs at points where the second derivative is equal to zero. An inflection point is where the concavity of the function graph changes from concave up to concave down, or vice versa.

To determine the inflection points, we follow these steps:

Find the first derivative f'(x).
Find the second derivative f''(x).
Solve for x such that f''(x) = 0.
Check intervals around the solutions for changes in the sign of f''(x) to confirm the presence of inflection points.

By applying these steps to the given function, you can find the x-values of the inflection points, denoted as (D), (E), and (F). Then you can classify the concavity of the function on the intervals around these points.