Question

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

Options

• 75

• 50

• 25

• 55

Answer 1

75

 

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

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

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

\[f'\left( x \right) = 0\]

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

\[ \Rightarrow 2 x^3 - 250 = 0\]

\[ \Rightarrow x^3 = 125\]

\[ \Rightarrow x = 5\]

\[\text { Now,} \]

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

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

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

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