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Answer :
F is not specified, we cannot determine the sign of f''(x) in this interval. The function f(x) = 12x^5 + 60x^4 - 240x^2 + 1 has inflection points at x=0, E, and F (assuming D is a typo).
To determine whether f(x) is concave up or concave down on each interval, we need to analyze the second derivative of f(x).
Taking the derivative of f(x) with respect to x, we get f'(x) = 60x^4 + 240x^3 - 480x.
Now, let's find the second derivative by taking the derivative of f'(x). The second derivative f''(x) = 240x^3 + 720x^2 - 480.
To find the intervals where f(x) is concave up or concave down, we need to solve f''(x) = 0.
Setting f''(x) equal to zero and factoring out 240, we have 240(x^3 + 3x^2 - 2) = 0.
Solving for x, we find that x = 0 is a critical point.
Now, let's analyze the sign of f''(x) in each interval:
1. (-∞, D): Since D is not specified, we cannot determine the sign of f''(x) in this interval.
2. (D, E): If D is the critical point at x = 0, we need to test a value between D and E. Let's say we choose x = 1. Plugging x = 1 into f''(x), we get f''(1) = 240(1^3 + 3(1)^2 - 2) = 240(2) = 480. Since the result is positive, f(x) is concave up on this interval.
3. (E, F): Without specific values for E and F, we cannot determine the sign of f''(x) in this interval.
4. (F, ∞): Since F is not specified, we cannot determine the sign of f''(x) in this interval.
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