Math  /  Calculus

QuestionConsider the function.(If an answer does not exist, enter DNE.) f(x)=(x4)2(x8)2f(x)=(x-4)^{2}(x-8)^{2} (a) Determine intervals where ff is increasing or decreasing. (Enter your answers using interval notation.) increasing \square decreasing \square (b) Determine the local minima and maxima of ff. (Enter your answers as comma-separated lists.) locations of local minima x=\quad x= \square locations of local maxima x=\quad x= \square (c) Determine intervals where ff is concave up or concave down. (Enter your answers using interval notation.) concave up \square concave down \square (d) Determine the locations of inflection points of ff. Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator. (Enter your answers as a comma-separated list.) x=x=\square

Studdy Solution
Based on our analysis, we can sketch the curve: - The function has local minima at x = 4 and x = 8 - It has a local maximum at x = 6 - It's concave up for x < 4.84 and x > 7.16 - It's concave down for 4.84 < x < 7.16 - It has inflection points at x ≈ 4.84 and x ≈ 7.16
The curve will look like a W shape, with the bottom of the W at x = 4 and x = 8, and the peak in the middle at x = 6.
Now, let's summarize our answers:
(a) increasing: (4, 6) ∪ (8, ∞) decreasing: (-∞, 4) ∪ (6, 8)
(b) locations of local minima: x = 4, 8 locations of local maxima: x = 6
(c) concave up: (-∞, 4.84) ∪ (7.16, ∞) concave down: (4.84, 7.16)
(d) x = 4.84, 7.16 (approximate values)

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