Find all the points of local maxima and local minima of the function
$f (x) = - \frac{3}{4} x^{4} - 8 x^{3} - \frac{45}{2} x^{2} + 105.$

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Solution

$f (x) = (\frac{- 3}{4}) x^{4} - 8 x^{3} - (\frac{45}{2}) x^{2} + 105 f^{'} (x) = (\frac{- 3}{4}) \times 4 x^{3} - 8 \times 3 x^{2} - (\frac{45}{2}) \times 2 x = - 3 x^{3} - 24 x^{2} - 45 x = - 3 [x^{2} + 8 x + 15] = - 3 [x^{2} + 5 x + 3 x + 15] = - 3 [x (x + 5) + 3 (x^{2} + 5)] = - 3 (x + 5) (x + 3) n o w f o r c r i t i c a l p o i n t s f^{'} (x) = 0 x = - 5 a n d x = - 3 ∴ a t x = - 5 i s a p o i n t o f m i n i m a ∴ a t x = - 3 i s a p o i n t o f m a x i m a .$

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