Question

The minimum value of $\left(x^2+\frac{250}{x}\right)$ is
(a) 75

(b) 50

(c) 25

(d) 55

Answer 1

(a) 75


$$\begin{gathered} \mathrm{Given}:f\left(x\right)=x^2+\frac{250}{x} \\ \Rightarrow f\text{'}\left(x\right)=2x-\frac{250}{x^2} \\ \mathrm{For}\mathrm{a}\mathrm{local}\mathrm{maxima}\mathrm{or}\mathrm{a}\mathrm{local}\mathrm{minima},\mathrm{we}\mathrm{must}\mathrm{have} \\ f\text{'}\left(x\right)=0 \\ \Rightarrow 2x-\frac{250}{x^2}=0 \\ \Rightarrow 2x^3-250=0 \\ \Rightarrow x^3=125 \\ \Rightarrow x=5 \\ \mathrm{Now}, \\ f\text{'}\text{'}\left(x\right)=2+\frac{500}{x^3} \\ \Rightarrow f\text{'}\text{'}\left(5\right)=2+\frac{500}{5^3}=\frac{750}{125}=6>0 \\ \mathrm{So},x=5\mathrm{is}\mathrm{a}\mathrm{local}\mathrm{minima}. \\ \therefore f\text{'}{\left(x\right)}_{\min =}{5}^2+\frac{250}{5}=\frac{375}{5}=75 \end{gathered}$$