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Q.

Let the function $f (x) = 2 x^{3} - (2 p - 7) x^{2} + 3 (2 p - 9) x - 6$ have a maximum for some value of $x < 0$ and a minima for some value of $x > 0$ . Then the set of all value of p is

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a

$(\frac{9}{2}, \infty)$

b

$(- \frac{9}{2}, \frac{9}{2})$

c

$(0, \frac{9}{2})$

d

$(- \infty, \frac{9}{2})$

answer is D.

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Detailed Solution

detailed_solution_thumbnail

$f (x) = 2 x^{3} + (2 p - 7) x^{2} + 3 (2 p - 9) x - 6$

$f^{'} (x) = 6 x^{2} + 2 x (2 p - 7) + 3 (2 p - 9)$

Since $α β < 0 \to 3 (2 p - 9) < 0,$

$2 p < 9 \Rightarrow p < \frac{9}{2}$
$p \in (- \infty, 9 / 2)$

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Let the function f(x)=2x3−(2p−7)x2+3(2p−9)x−6 have a maximum for some value of x<0 and a minima for some value of x>0. Then the set of all value of p is