Question Number 56900 by bshahid010@gmail.com last updated on 26/Mar/19
Answered by tanmay.chaudhury50@gmail.com last updated on 26/Mar/19
$${f}\left({x}\right)=\mid\left({x}−{a}\right)^{\mathrm{3}} \mid+\mid\left({x}−{b}\right)^{\mathrm{3}} \mid \\ $$$$\:\:\:\:\:\:\:\:\:\:=\left({x}−{a}\right)^{\mathrm{3}} +\left({x}−{b}\right)^{\mathrm{3}} \:\:\:\:\:{when}\:{x}>{b} \\ $$$$\:\:\:\:\:\:\:\:\:\:=\left({b}−{a}\right)^{\mathrm{3}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{when}\:{x}={b} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\left({x}−{a}\right)^{\mathrm{3}} −\left({x}−{b}\right)^{\mathrm{3}} \:\:\:\:\:\:\:{when}\:{b}>{x}>{a} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=−\left({a}−{b}\right)^{\mathrm{3}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{when}\:{x}={a} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\left({b}−{a}\right)^{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=−\left({x}−{a}\right)^{\mathrm{3}} −\left({x}−{b}\right)^{\mathrm{3}} \:\:{when}\:{x}<{a} \\ $$$$ \\ $$$${when}\:{x}>{b} \\ $$$${f}\left({x}\right)=\left({x}−{a}\right)^{\mathrm{3}} +\left({x}−{b}\right)^{\mathrm{3}} \\ $$$$\frac{{df}}{{dx}}=\mathrm{3}\left({x}−{a}\right)^{\mathrm{2}} +\mathrm{3}\left({x}−{b}\right)^{\mathrm{2}} \\ $$$$\frac{{df}}{{dx}}\neq\mathrm{0}\:\:\:\:{as}\:{x}>{b} \\ $$$${but}\:{for}\:{max}/{min}\:\frac{{df}}{{dx}}=\mathrm{0} \\ $$$$\frac{{d}^{\mathrm{2}} {f}}{{dx}^{\mathrm{2}} }=\mathrm{6}\left({x}−{a}\right)+\mathrm{6}\left({x}−{b}\right) \\ $$$${so}\:{no}\:{max}/{min}\:{value}\:{at}\:{x}>{b} \\ $$$$ \\ $$$${when}\:{b}>{x}>{a} \\ $$$${f}\left({x}\right)=\left({x}−{a}\right)^{\mathrm{3}} −\left({x}−{b}\right)^{\mathrm{3}} \\ $$$$\frac{{df}}{{dx}}=\mathrm{3}\left({x}−{a}\right)^{\mathrm{2}} −\mathrm{3}\left({x}−{b}\right)^{\mathrm{2}} \\ $$$$\frac{{df}}{{dx}}=\mathrm{0}\:\:{at}\:{x}=\frac{{a}+{b}}{\mathrm{2}} \\ $$$$\frac{{d}^{\mathrm{2}} {f}}{{dx}^{\mathrm{2}} }=\mathrm{6}\left({x}−{a}\right)−\mathrm{6}\left({x}−{b}\right)=\mathrm{6}\left({b}−{a}\right)>\mathrm{0} \\ $$$${so}\:{at}\:{x}=\frac{{a}+{b}}{\mathrm{2}}\:{i},{e}\:{f}\left(\frac{{a}+{b}}{\mathrm{2}}\right)={minimum} \\ $$$${f}\left(\frac{{a}+{b}}{\mathrm{2}}\right)=\left(\frac{{a}+{b}}{\mathrm{2}}−{a}\right)^{\mathrm{3}} −\left(\frac{{a}+{b}}{\mathrm{2}}−{b}\right)^{\mathrm{3}} \\ $$$$=\left(\frac{{b}−{a}}{\mathrm{2}}\right)^{\mathrm{3}} −\left(\frac{{a}−{b}}{\mathrm{2}}\right)^{\mathrm{3}} \\ $$$$=\mathrm{2}×\left(\frac{{b}−{a}}{\mathrm{2}}\right)^{\mathrm{3}} \leftarrow{this}\:{is}\:{the}\:{answer}\:\left({minimum}\:{value}\right) \\ $$$${when}\:{x}<{a} \\ $$$${f}\left({x}\right)=−\left({x}−{a}\right)^{\mathrm{3}} −\left({x}−{b}\right)^{\mathrm{3}} \\ $$$$\frac{{df}}{{dx}}=−\mathrm{3}\left[\left({x}−{a}\right)^{\mathrm{2}} +\left({x}−{b}\right)^{\mathrm{2}} \right] \\ $$$${but}\:\frac{{df}}{{dx}}\neq\mathrm{0}\:\: \\ $$$${so}\:{no}\:{min}/{max}\:{when}\:{x}<{a} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$