To examine the function $f (x) = 2 x^{3} - 15 x^{2} + 36 x + 10$ for maxima and minima, if any. Also find the maximum and minimum value.

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Solution

$f (x) = 2 x^{3} - 15 x^{2} + 36 x + 10 \to (i) d i f f e r e n t i a t e w . r t o x \Rightarrow f^{'} (x) = 6 x^{2} - 30 x + 36 \Rightarrow f^{'} (x) = 0 \Rightarrow 6 x^{2} - 30 x + 36 = 0 \Rightarrow x^{2} - 5 x + 6 = 0 \Rightarrow x^{2} - 3 x - 2 x + 6 = 0 \Rightarrow x (x - 3) - 2 (x - 3) = 0 \Rightarrow x = 3, x = 2 a g a i n d i f f e r e n t i a t e w . r t o x f^{'} (x) = 12 x - 30 \to (i i) P u t x = 2 \Rightarrow f^{'} (2) = 24 - 30 = - 60, x = 2 p o i n t m a x i m u m a n d m i n i m u m v a l u e i s f (2) = 2 \times 2^{3} - 15 \times 2^{2} + 36 \times 2 + 10 \Rightarrow f (2) = 16 - 60 + 71 + 10 = 38 a n d p u t x = 3 i n e q^{n} (i i) \Rightarrow f^{'} (3) = 36 - 30 = 6 I t i s p o i n t m a x i m u m a n d m i n i m u m v a l u e i s f (3) = 2 \times 27 - 15 \times 9 + 36 \times 3 + 10 = 37$

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