Answer :
Final answer:
To find the intervals of concavity for the function f(x) = -2x^6 + 9x^5 + 50x^4 + 60x + 60, we find the second derivative f''(x) = -60x^4 + 180x^3 + 400x^2 + 60. By analyzing the sign of f''(x), we determine f(x) is concave up in the intervals (-∞, -2) and (0, 1), and concave down in the interval (-2, 0).
Explanation:
The concavity of a function can be determined by analyzing the concavity of its derivative, which is the second derivative of the function. To find the intervals of concavity for the given function f(x) = -2x^6 + 9x^5 + 50x^4 + 60x + 60, we need to find its second derivative. Taking the derivative twice, we get f''(x) = -60x^4 + 180x^3 + 400x^2 + 60.
To find the intervals of concavity, we need to analyze the sign of f''(x). A positive value indicates concave up, while a negative value indicates concave down. To find the critical points of f''(x), we solve f''(x) = 0. By factoring out common terms and using the zero-product property, we find the critical points are x = -2, x = 0, and x = 1.
Now, we choose test points to determine the concavity in the intervals. We can pick any number from each interval and evaluate f''(x) at that point. Testing in the intervals (-∞, -2), (-2, 0), (0, 1), and (1, ∞), we find that f''(x) is positive in the intervals (-∞, -2) and (0, 1), indicating concave up, and f''(x) is negative in the interval (-2, 0), indicating concave down.
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