6. Application of derivatives — ગણિત ધોરણ 12 સાયન્સ — Question

$f' (x) = 8e^{2x} - 18e^{-2x}$ 

હવે $f$  એ જે બિંદુએ ન્યનતમ હોય તે બિંદુએ $f' (x) = 0$

$\therefore \,\,8{e^{2x}}\, - \,18{e^{ - 2x}}\, = \,\,0\,\,\,\,\,$

$\therefore \,\,8{e^{2x}}\,\, = \,\,{18^{ - 2x}}\,\,\,\,\,$

$\therefore \,\,{\left( {{e^{2x}}} \right)^2}\,\, = \,\,\frac{{18}}{8}\,\,\,\,\,$

$\therefore \,\,{e^{2x}}\,\, = \, \pm \,\frac{3}{2}$

વળી ${\bf{f''}}{\rm{(x)  =  16}}{{\rm{e}}^{{\rm{2x}}}}{\rm{  +  36}}{{\rm{e}}^{{\rm{ - 2x}}}}{\rm{ }} = \,\,16{e^{2x}}\, + \,\frac{{36}}{{{e^{2x}}}}$

$\therefore \,\,{e^{2x}}\,\, = \,\,\frac{3}{2}$ માટે ${\bf{f''}}{\rm{(x)  >  0 }}$

$\therefore \,\,{e^{2x}}\,\, = \,\,\frac{3}{2}\,\,$ આગળ ${f}{\rm{ }}$ ન્યુનતમ છે અને આ ન્યુનતમ  કિમત 

$ = \,\,4\left( {\frac{3}{2}} \right)\,\, + \,9\left( {\frac{2}{3}} \right)\,\, = \,\,6\, + \,\,6\,\, = \,\,12$