Answer :
Final answer:
The function f(x)=x⁵-5 x⁴+23 is both increasing and concave up on the interval (4, ∞).
Explanation:
We first need to find the first derivative of the function, f'(x) = 5x⁴ - 20x³. A function is increasing when its derivative is positive. To find where f'(x) is positive, we find the critical points by setting f'(x) = 0, that gives us x = 0 and x = 4. Hence, the function is increasing in the interval (-∞, 0) union (4, ∞).
Next, we find the second derivative, f''(x) = 20x³ - 60x². A function is concave up when its second derivative is positive. Setting f''(x) = 0 gives us x = 0 and x = 3. Thus, the function is concave up on the interval (0, 3) and (3, ∞).
The intersection of the intervals where the function is increasing and is concave up gives the answer: (4, ∞).
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