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Question
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
Options
75
50
25
55
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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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