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Question
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
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Solution
f(x) = 2x3 - 15x2 + 36x + 1
∴ f'(x) = 6x2 - 30x + 36
= 6(x2 - 5x + 6)
= 6(x - 3)(x - 2)
f(x) is an increasing function, if f'(x) > 0
∴ 6(x - 3)(x - 2) > 0
∴ (x - 3)(x - 2) > 0
ab > 0 ⇔ a > 0 and b > 0 or a < 0 or b < 0
∴ Either (x – 3) > 0 and (x – 2) > 0 or
(x – 3) < 0 and (x – 2) < 0
Case 1: x – 3 > 0 and x – 2 > 0
∴ x > 3 and x > 2
∴ x > 3
Case 2: x – 3 < 0 and x – 2 < 0
∴ x < 3 and x < 2
∴ x < 2
Thus, f(x) is an increasing function for x < 2 or x > 3, i.e., (- ∞, 2) ∪ (3, ∞)
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