calc
posted by sarah .
let f be function given by f(x)= Ln(x)/x for all x> 0. the dervative of f is given by f'(x)= (1  Ln(x))/x squared.
a) write equation for the line tangent to the graph of f at x=e squared
b) Find the xcoordinate of the critical point of f. Determine wheter this point is a relative minimum, a relative maximum or neither for the function f.
c) the graph of the function f has exactly one point of inflation. find the x coordinates of this point.
(d) find lim f(x) x to the 0+

calc 
Reiny
a) when x=e^2, y = lne^2/e^2 = 2/e^2 and
y' = (1lne^2)/e^4 = 1/e^4
so the equation is x + (e^4)y = c
but (e^2,2/e^2) lies on it, so
e^2 + e^4(2/e^2) = c
c = 3e^2
the equation is x + (e^4)y = 3e^2
( I am using a quick way to find the equation of a line based on the fact that if the slope is A/B, then the equation is Ax + By = C ) 
calc 
Reiny
b) set f'(x) = 0
(1 lnx)/x^2 = 0
1  ln x = 0
ln x = 1
x = e then y = ln e/e = 1/e
the critical point is (1,1/e)
To determine if it is a max/min I will do c) first 
calc 
Reiny
c) for f''(x) I got
(3 + 2lnx)/x^3
setting that equal to zero ...
2lnx = 3
lnx = 3/2
x = 4.4817 and y = .3347
so there is a point of inflection at (4.48,0.33)
Back to b)
when you plug x=e^2 into f'' you get
(3+ 2lne^2)/e^6 which is positive.
Therefore the critical point from b) is a minimum point of the function.
Respond to this Question
Similar Questions

Calc
Find an equation of the tangent line to the graph of the function at the given point. 1 + ln xy = e^(xy), (1,1) 
Calculus  Functions?
#1. A cubic polynomial function f is defined by f(x) = 4x^3 +ax^2 + bx + k where a, b and k are constants. The function f has a local minimum at x = 1, and the graph of f has a point of inflection at x= 2 a.) Find the values of a … 
Calc I
I really need assistance!! I am to write an equation so that I can graph it. I do not understand how to graph OR write the equation for: x^2y^2=16 for a tangent and normal line? 
calculous
3.Given the function f defined by f(x)=2x^33x^212x+20 a.Find the zeros of f b.Write an equation of the line perpendicular to the graph of f at x = 0 c. Find the x and y coordinates of all points on the graph of f where the line tangent … 
Calculus
1. Given the function f defined by f(x) = x^3x^24X+4 a. Find the zeros of f b. Write an equation of the line tangent to the graph of f at x = 1 c. The point (a, b) is on the graph of f and the line tangent to the graph at (a, b) … 
Calculus
1. a) For the Function and point below , Find f’(a). b) Determine the equation of the line tangent to the graph of f at (a,f(a)) for the given value of f(x) = 4x2+2x, a =1 F’(a) = y = 2. For the function find f’ using the definition … 
Calculus
1. a) For the Function and point below , Find f’(a). b) Determine the equation of the line tangent to the graph of f at (a,f(a)) for the given value of f(x) = 4x2+2x, a =1 F’(a) = y = 2. For the function find f’ using the definition … 
Calc
Write the equation of the line tangent to the graph of the function at the indicated point. As a check, graph both the function and the tangent line you found to see whether it looks correct. (1/x^3 x)^2 at x= 2 
Calculus
The function g is defined for x>0 with g(1)=2, g'(x)=sin(x+1/x), and g"(x)=(11/x^2)cos(x+1/x). A. Find all values of x in the interval 0.12<=x<=1 at which the graph of g has a horizontal tangent line. B. On what subintervals … 
math, a.p. calc
Let g be a function that is defined for all x, x ≠ 2, such that g(3)=4 and the derivative of g is g′(x)= x^2–16/x−2, with x≠2. Write an equation for the tangent line to the graph of g at the point where x …