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# Let F(X) = X3+3x2 − 9x+2. Then, F(X) Has (A) a Maximum at X = 1 (B) a Minimum at X = 1 (C) Neither a Maximum Nor a Minimum at X = − 3 (D) None of These - Mathematics

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#### Question

Let f(x) = x3+3x$-$ 9x+2. Then, f(x) has _________________ .

##### Options
• a maximum at x = 1

• a minimum at x = 1

• neither a maximum nor a minimum at x = - 3

• none of these

#### Solution

$\text { a minimum at x = 1}$

$\text { Given }: f\left( x \right) = x^3 + 3 x^2 - 9x + 2$

$\Rightarrow f'\left( x \right) = 3 x^2 + 6x - 9$

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

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

$\Rightarrow 3 x^2 + 6x - 9 = 0$

$\Rightarrow x^2 + 2x - 3 = 0$

$\Rightarrow \left( x + 3 \right)\left( x - 1 \right) = 0$

$\Rightarrow x = - 3, 1$

$\text { Now,}$

$f''\left( x \right) = 6x + 6$

$\Rightarrow f''\left( 1 \right) = 6 + 6 = 12 > 0$

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

$\text { Also },$

$f''\left( - 3 \right) = - 18 + 6 = - 12 < 0$

$\text { So, x = - 3 is a local maxima } .$

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Let F(X) = X3+3x2 − 9x+2. Then, F(X) Has (A) a Maximum at X = 1 (B) a Minimum at X = 1 (C) Neither a Maximum Nor a Minimum at X = − 3 (D) None of These Concept: Graph of Maxima and Minima.
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