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Question
Choose the correct alternative.
If f(x) = 3x3 - 9x2 - 27x + 15 then
Options
f has maximum value 66
f has minimum value 30
f has maxima at x = –1
f has minima at x = –1
Solution
f has maxima at x = –1
Explanation:
f(x) = 3x3 - 9x2 - 27x + 15
∴ f'(x) = 9x2 - 18x - 27
∴ f''(x) = 18x - 18
Consider, f '(x) = 0
∴ 9x2 - 18x - 27 = 0
∴ x2 - 2x- 3 = 0
∴ (x – 3) (x + 1) = 0
∴ x = 3 or x = – 1
For x = 3, f ''(x) = 18(3) – 18 = 36 > 0
∴ f(x) has minimum value at x = 3
∴ Minimum value = f(3) = – 66
For x = – 1, f ''(x) = 18(–1) – 18 = – 36 < 0
∴ f(x) has maximum value at x = –1
∴ Maximum value = f(–1) = 30.
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