Question

The number which exceeds its square by the greatest possible quantity is
(a) $\frac{1}{2}$

(b) $\frac{1}{4}$

(c) $\frac{3}{4}$

(d) none of these

Answer 1

$$\begin{gathered} \left(a\right)\frac{1}{2} \\ \mathrm{Let}\mathrm{the}\mathrm{required}\mathrm{number}\mathrm{be}x.\mathrm{Then}, \\ f\left(x\right)=x-x^2 \\ \Rightarrow f\text{'}\left(x\right)=1-2x \\ \mathrm{For}\mathrm{a}\mathrm{local}\mathrm{maxima}\mathrm{or}\mathrm{a}\mathrm{local}\mathrm{minima},\mathrm{we}\mathrm{must}\mathrm{have} \\ f\text{'}\left(x\right)=0 \\ \Rightarrow 1-2x=0 \\ \Rightarrow 2x=1 \\ \Rightarrow x=\frac{1}{2} \\ \mathrm{Now}, \\ f\text{'}\text{'}\left(x\right)=-2<0 \\ \mathrm{So},x=\frac{1}{2}\mathrm{is}\mathrm{a}\mathrm{local}\mathrm{maxima}. \\ \mathrm{Hence},\mathrm{the}\mathrm{required}\mathrm{number}\mathrm{is}\frac{1}{2}. \end{gathered}$$