Răspuns:
f(x)=x²lnx/2-3x²/4
f `(x)=2xlnx/2+x²/2x-6x/4=
xlnx+x/2-3x/2=
xlnx-2x/2=
xlnx-x
Determini punctul de extrem f `(x)=0
xlnx-x=0
x(lnx-1)=0=>
1=0∉(0,+∞)
lnx-1=0
lnx-1=0
lnx=1
lnx=1
x=e
pt x<e lnx-1<0
Pt x>e lnx-1>0
=>x=e mimimull functiei f(x)
f(e)=e²lne/2-3e²/4=
e²/2-3e²/4=
2e²/4-3e²/4=-e²/4
Acesta fiind punct de minim =>
f(x)≥ -e²/4 ∀x>0
Explicație pas cu pas:
