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# Let F(X) = 2x3 − 3x2 − 12x + 5 on [ 2, 4]. the Relative Maximum Occurs at X = (A) − 2 (B) − 1 (C) 2 (D) 4 - Mathematics

#### Question

Let f(x) = 2x3$-$ 3x2$-$ 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .

• -2

• -1

• 2

• 4

#### Solution

2

$\text { Given }: f\left( x \right) = 2 x^3 - 3 x^2 - 12x + 5$

$\Rightarrow f'\left( x \right) = 6 x^2 - 6x - 12$

$\text { For a local maxima or a local minima, we must have }$

$f'\left( x \right) = 0$

$\Rightarrow 6 x^2 - 6x - 12 = 0$

$\Rightarrow x^2 - x - 2 = 0$

$\Rightarrow \left( x - 2 \right)\left( x + 1 \right) = 0$

$\Rightarrow x = 2, - 1$

$\text{ Now, }$

$f''\left( x \right) = 12x - 6$

$\Rightarrow f''\left( - 1 \right) = - 12 - 6 = - 18 < 0$

$\text { So, x = 1 is a local maxima } .$

$\text { Also },$

$f''\left( 2 \right) = 24 - 6 = 18 > 0$

$\text { So, x = 2 is a local minima } .$

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