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Question
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
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Solution
f(x) = x3 – 9x2 + 24x + 12
∴ f'(x) = `d/dx(x^3 - 9x^2 + 24x + 12)`
= 3x2 – 9 x 2x + 24 x 1 + 0
= 3x2 – 18x + 24
= 3(x2 – 6x + 8)
f is strictly decreasing if f'(x) < 0
i.e. if 3(x2 – 6x + 8) < 0
i.e. if x2 – 6x + 8 < 0
i.e. if x2 – 6x < – 8
i.e. if x2 – 6x + 9 < – 8 + 9
i.e. if (x – 3)2 < 1
i.e. if – 1 < x – 3 < 1
i.e. if – 1 + 3 < x – 3 + 3 < 1 + 3
i.e. if 2 < x < 4
i.e., if x ∈ (2, 4)
∴ f is strictly decreasing if x ∈ (2, 4).
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