Question Number 13075 by Joel577 last updated on 13/May/17
$${S}\left({x}\right)\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{49}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{AP} \\ $$$$\mathrm{The}\:\mathrm{first}\:\mathrm{term}\:\mathrm{is}\:\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{3}} \:\mathrm{and}\:\mathrm{the}\:\mathrm{difference} \\ $$$$\mathrm{is}\:\left(\mathrm{7}\:−\:{x}\right) \\ $$$$\mathrm{If}\:{S}\left({x}\right)\:\mathrm{maximum},\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{10}^{\mathrm{th}} \:\mathrm{term}\:\mathrm{is}\:... \\ $$
Commented by ajfour last updated on 13/May/17
![local maximum ? x=−4 then, T_(10) =(((−4)^3 )/2)+9[7−(−4)] = −32+99 =67 .](Q13077.png)
$${local}\:{maximum}\:?\:\:{x}=−\mathrm{4}\:\:\:{then}, \\ $$$$\:{T}_{\mathrm{10}} =\frac{\left(−\mathrm{4}\right)^{\mathrm{3}} }{\mathrm{2}}+\mathrm{9}\left[\mathrm{7}−\left(−\mathrm{4}\right)\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:−\mathrm{32}+\mathrm{99}\:=\mathrm{67}\:. \\ $$
Commented by Joel577 last updated on 13/May/17
$$\mathrm{How}\:\mathrm{did}\:\mathrm{u}\:\mathrm{get}\:{x}\:=\:−\mathrm{4}\:? \\ $$
Commented by ajfour last updated on 13/May/17
$${S}\left({x}\right)=\frac{\mathrm{49}}{\mathrm{2}}\left\{\mathrm{2}\left(\frac{{x}^{\mathrm{3}} }{\mathrm{2}}\right)+\mathrm{48}\left(\mathrm{7}−{x}\right)\right\} \\ $$$$\frac{{dS}}{{dx}}=\frac{\mathrm{49}}{\mathrm{2}}\left\{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{48}\right\}=\mathrm{0} \\ $$$$\Rightarrow\:{x}^{\mathrm{2}} =\mathrm{16} \\ $$$$\:\:\:\:\:{x}=\pm\mathrm{4}\:. \\ $$
Commented by ajfour last updated on 13/May/17
Commented by Joel577 last updated on 13/May/17
$${thank}\:{you}\:{very}\:{much} \\ $$