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Question
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
Options
0
12
16
32
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Solution
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is 12.
Explanation:
Given that y = –x3 + 3x2 + 9x – 27
`"dy"/'dx"` = – 3x2 + 6x + 9
∴ Slope of the given curve,
m = – 3x2 + 6x + 9 ....`("dy"/"dx" = "m")`
`"dm"/"dx"` = –6x + 6
For local maxima and local minima, `"dm"/"dx"` = 0
∴ – 6x + 6 = 0
⇒ x = 1
Now `("d"^2"m")/("dx"^2)` = = – 6 < 0 maxima
∴ Maximum value of the slope at x = 1 is
`"m"_(x = 1)` = – 3(1)2 + 6(1) + 9
= – 3 + 6 + 9
= 12
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