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Assertion A: ...

Question

Assertion (A): The function $f (x) = 2 x^{3} - 3 x^{2_{-}} 12 x + 8$ has minimum value -12 at x = 2
Reason (R): For the fucntion $f (x) = 2 x^{3} - 3 x^{2} - 12 x + 8, f^{'} (2) = 0$ and $f^{''} (2) > 0$

Both A and R are true and R is the correct explanation of A

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Both A and R are true and R is not the correct explanation of A

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A is true and R is false

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A is false and R is true

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Solution

The correct option is D Both A and R are true and R is the correct explanation of A
$f^{'} (x) = 6 x^{2} + 6 x - 12$
$= 6 (x^{2} - x - 2)$
$= 6 (x - 2) (x + 1)$
$f^{''} (x) < 0$ If $- 1 < x < 2$
$f^{'} (x) > 0$ If $x > 2$ or $x < - 1$
$f (x)$ is max at $x = 2$
$f (2) = - 12$
$f^{'} (2) = 0$
$f^{'} (x) = 6 (2 x - 1)$
$f^{''} (2) > 0$

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