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The Minimum Value of ( X 2 + 250 X ) is (A) 75 (B) 50 (C) 25 (D) 55 - CBSE (Science) Class 12 - Mathematics

Question

The minimum value of $\left( x^2 + \frac{250}{x} \right)$ is __________ .

• 75

• 50

• 25

• 55

Solution

75

$\text { Given }: f\left( x \right) = x^2 + \frac{250}{x}$

$\Rightarrow f'\left( x \right) = 2x - \frac{250}{x^2}$

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

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

$\Rightarrow 2x - \frac{250}{x^2} = 0$

$\Rightarrow 2 x^3 - 250 = 0$

$\Rightarrow x^3 = 125$

$\Rightarrow x = 5$

$\text { Now,}$

$f''\left( x \right) = 2 + \frac{500}{x^3}$

$\Rightarrow f''\left( 5 \right) = 2 + \frac{500}{5^3} = \frac{750}{125} = 6 > 0$

$\text { So, x = 5 is a local minima } .$

$\therefore f' \left( x \right)_\min = 5^2 + \frac{250}{5} = \frac{375}{5} = 75$

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Solution The Minimum Value of ( X 2 + 250 X ) is (A) 75 (B) 50 (C) 25 (D) 55 Concept: Graph of Maxima and Minima.
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