Consider the function [tex]f(x) = 12x^5 + 45x^4 - 80x^3 + 1[/tex].

The function [tex]f(x)[/tex] has inflection points at [tex]x = D, E,[/tex] and [tex]F[/tex] (reading from left to right), where [tex]D[/tex] is, [tex]E[/tex] is, and [tex]F[/tex] is.

For each of the following intervals, determine whether [tex]f(x)[/tex] is concave up or concave down.

To determine the concavity of the function f(x), we need to find the sign of its second derivative. If the second derivative is positive, the function is concave up, and if it is negative, the function is concave down.

Explanation:

f(x) = 12ˣ⁵+45ˣ⁴−80ˣ³ +1 has inflection points at x=D, E, and F. To determine whether f(x) is concave up or concave down in different intervals, we need to find the sign of its second derivative. The second derivative, f''(x), represents the concavity of the function. If f''(x) > 0, the function is concave up, and if f''(x) < 0, the function is concave down.

To find the second derivative, we differentiate f(x) twice:

f'(x) = 60ˣ⁴ + 180ˣ³ - 240ˣ²

f''(x) = 240ˣ³ + 540ˣ² - 480ˣ

Now, we can evaluate f''(x) at the given inflection points

f''(D), f''(E), and f''(F)

If f''(D) > 0, f(x) is concave up at x=D. If f''(E) < 0, f(x) is concave down at x=E. Similarly, if f''(F) > 0, f(x) is concave up at x=F.