Find the intervals in which the function $f (x)$ is (i) increasing, (ii) decreasing
$f (x) = 2 x^{3} - 9 x^{2} + 12 x + 15$

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Solution

$\begin{matrix} L e t f (x) = 2 x^{3} - 9 x^{2} + 12 x + 15 f^{'} (x) = 6 x^{2} - 18 x + 12 = 6 (x^{2} - 3 x + 2) = 6 (x - 1) (x - 2) f^{'} (x) = 0 g i v e s u s 6 (x - 1) (x - 2) = 0 \Rightarrow x = 1, 2 T h e p o i n t s x = 1, 2 d i v i d e t h e r e a l l l i n e i n t o t h r e e i n t e r v a l s (- \infty, 1), (1, 2), (2, \infty) 1. I n t h e i n t e r v a l (- \infty, 1) f^{'} (x) > 0 ∴ f (x) i s i n c r e a sin g i n (- \infty, 1) 2. I n t h e i n t e r v a l (1, 2), f^{'} (x) < 0 ∴ f (x) i s d e c r e a sin g i n (1, 2) 3. I n t h e i n t e r v a l (2, \infty) f^{'} (x) > 0 ∴ f (x) i s i n c r e a s i n g i n (2, \infty) . \end{matrix}$

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