Consider the function \( f(x) = 12x^5 + 45x^4 - 80x^3 + 1 \).

It has inflection points at \( x = D \), \( x = E \), and \( x = F \).

For each of the following intervals, determine whether \( f \) is concave up or concave down.

To assess the concavity of f(x), calculate the second derivative and check its sign within the intervals between the inflection points D, E, and F. The function is concave up where f''(x) > 0 and concave down where f''(x) < 0.

Explanation:

To determine whether the function f(x) = 12x⁵ + 45x⁴ - 80x³ + 1 is concave up or down, we need to analyze the sign of its second derivative, f''(x). An inflection point occurs where the second derivative is zero and the concavity changes.

At points where f''(x) > 0, the function is concave up, resembling a cup that holds water. Conversely, when f''(x) < 0, the function is concave down, resembling a cup turned upside down. In intervals between inflection points (where the second derivative does not change sign), the concavity will remain consistent. To find out the concavity between the inflection points D, E, and F, you would calculate f''(x) and test points in each interval to see whether f''(x) is positive or negative.

Consider the function \( f(x) = 12x^5 + 45x^4 - 80x^3 + 1 \).It has inflection points at \( x = D \), \( x = E \), and \( x = F \).For each of the following intervals, determine whether \( f \) is concave up or concave down.

Answer :

Final answer:

Explanation:

Other Questions

Consider the function \( f(x) = 12x^5 + 45x^4 - 80x^3 + 1 \).

It has inflection points at \( x = D \), \( x = E \), and \( x = F \).

For each of the following intervals, determine whether \( f \) is concave up or concave down.