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Question
Find all the points of local maxima and local minima of the function f(x) = `- 3/4 x^4 - 8x^3 - 45/2 x^2 + 105`
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Solution
f′(x) = –3x3 – 24x2 – 45x
= – 3x(x2 + 8x + 15)
= – 3x(x + 5)(x + 3)
f′(x) = 0
⇒ x = –5, x = –3, x = 0
f″(x) = –9x2 – 48x – 45
= –3(3x2 + 16x + 15)
f″(0) = – 45 < 0. Therefore, x = 0 is point of local maxima
f″(–3) = 18 > 0. Therefore, x = –3 is point of local minima
f″(–5) = –30 < 0. Therefore x = –5 is point of local maxima.
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