Find the greatest and least values of the following function of the indicated intervals:
$f (x) = 2 x^{3} - 3 x^{2} - 12 x + 1$ on [-2, 5/2];

g r e a t e s t = 12, l e a s t = - 17

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g r e a t e s t = 8, l e a s t = - 21

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g r e a t e s t = 8, l e a s t = - 19

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g r e a t e s t = 12, l e a s t = - 19

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Solution

The correct option is C $g r e a t e s t = 8, l e a s t = - 19$
$f (x) = 2 x^{3} - 3 x^{2} - 12 x + 1$
$\Rightarrow f^{'} (x) = 6 x^{2} - 6 x - 12$
For maxima or minima,
$f^{'} (x) = 0$
$\Rightarrow 6 (x^{2} - x - 2) = 0$
$\Rightarrow x = - 1, 2$
Now, $x = - 1, 2 \in [- 2, \frac{5}{2}]$
So, $f (- 2) = - 3$
$f (- 1) = 8$
$f (2) = - 19$
$f (\frac{5}{2}) = - \frac{33}{2}$
Hence,the greatest value is 8 and least value is -19

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