[tex]f(x)=\ln x-\frac{2(x-1)}{x}[/tex]
a)
[tex]f'(x)=\frac{1}{x}-\frac{2x-2(x-1)}{x^2} =\frac{x-2x+2x-2}{x^2}=\frac{x-2}{x^2}[/tex]
b)
Doua drepte sunt paralele daca pantele lor sunt egale
[tex]m_1=f'(a)\\\\m_2=-1[/tex]
f'(a)=-1
[tex]\frac{a-2}{a^2} =-1\\\\-a^2=a-2\\\\a^2+a-2=0\\\\\Delta=1+8=9\\\\a_1=\frac{-1+\sqrt{9} }{2}=1 \\\\a_2=\frac{-1-\sqrt{9} }{2}=-2 < 0\ Nu[/tex]
c)
Monotonia functiei f
Pe intervalul (0,2) x-2<0 ⇒ f'(x)<0⇒ f este descrescatoare pe (0,2)
[tex]1 < \frac{\pi}{2}[/tex] ⇒
[tex]f(1) > f(\frac{\pi}{2} )\\\\f(1)=0\\\\[/tex]⇒[tex]f(\frac{\pi}{2}) < 0[/tex]
Un alt exercitiu gasesti aici: https://brainly.ro/tema/133500
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