let f be the function defined over { 0 , infinety ) as integration of f(x) = x)lnx-1)

[kong]

let f be the function defined over { 0 , infinety ) as integration of f(x) = x)lnx-1)^2 for x>0 and let (c) be its representative curve , we admit that f is continous at x=0

a ) determine lim f(x) / x approaching 0

b) find lim f(x) approaching infinity and lim f(x) /x approaching infinity

c ) for x > 0, verify that derivative d/dx f(x) = (lnx)^2 - 1 and set up the table of variation of f

d) show that (c) has a point of inflection i and write an equation of (T) the tangent at I to (c)

e) the line m with equation y=x intersects (c) at three points 0 , i and j calculate the coordinates of j

f) plot (t) and (c)

g) show that the function f defined on ( 0, infinity ) as f(x)= x^2/2 ((lnx)^2 - 3lnx + 5/2 ) is an antiderivative of f

h ) deduce the area of the region bounded by the x-axis the tangennt ( t) and the curve ( c)

i ) for all x in the interval { e , infinity ) prove that f has an inverse function f^-1 and plot the representative curve of f^-1 in the same grid of part f)

given: let ( dm) be the line with equation y=mx intersects the curve ( c ) at three distinct points o, m and n

j ) calculate , in terms of m the points m and n

k ) denote by p the point on ( dm) with x-coordinate x= e , prove that om * on = op^2

 

[Deleted member 4993]

let f be the function defined over { 0 , infinety ) as integration of f(x) = x)lnx-1)^2 for x>0 and let (c) be its representative curve , we admit that f is continous at x=0

a ) determine lim f(x) / x approaching 0

b) find lim f(x) approaching infinity and lim f(x) /x approaching infinity

c ) for x > 0, verify that derivative d/dx f(x) = (lnx)^2 - 1 and set up the table of variation of f

d) show that (c) has a point of inflection i and write an equation of (T) the tangent at I to (c)

e) the line m with equation y=x intersects (c) at three points 0 , i and j calculate the coordinates of j

f) plot (t) and (c)

g) show that the function f defined on ( 0, infinity ) as f(x)= x^2/2 ((lnx)^2 - 3lnx + 5/2 ) is an antiderivative of f

h ) deduce the area of the region bounded by the x-axis the tangennt ( t) and the curve ( c)

i ) for all x in the interval { e , infinity ) prove that f has an inverse function f^-1 and plot the representative curve of f^-1 in the same grid of part f)

given: let ( dm) be the line with equation y=mx intersects the curve ( c ) at three distinct points o, m and n

j ) calculate , in terms of m the points m and n

k ) denote by p the point on ( dm) with x-coordinate x= e , prove that om * on = op^2

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