# calc

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 x-coordinate 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+

1. 👍 0
2. 👎 0
3. 👁 392
1. a) when x=e^2, y = lne^2/e^2 = 2/e^2 and

y' = (1-lne^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 )

1. 👍 0
2. 👎 0
2. 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

1. 👍 0
2. 👎 0
3. 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.

1. 👍 0
2. 👎 0

## Similar Questions

1. ### Algebra 2

The graph of an absolute value function opens up and has a vertex of (0, -3). The domain of the function is . The range of the function is

2. ### Math

compare the parent function f(x)=x^2 to the quadractic function f(x)=-2x^2-6. the 6 in the function does which of the following? a.it makes the graph narrower than the parent function. b.it makes the graph wider than the parent

3. ### math

The cost of producing q articles is given by the function C=f(q)=100+2q. (A) Find the formula for the inverse function. (B) Explain in practical terms what the inverse function tells you. I am pretty sure the answer to A is

4. ### Math

Assume it costs Microsoft \$4,300 to manufacture 7 Xbox 360s and \$8,290 to manufacture 14. Obtain the corresponding linear cost function. C(x)= (what formula do I use to get the cost function, because I can't get it) What was the

1. ### Calculus. I need help!

Write the composite function in the form f(g(x)). [Identify the inner function u=g(x) and the outer function y=f(u).] Then find the derivative dy/dx. y=√sinx √ is square root.

2. ### Algebra 2 check my last answer?!

A cosine function has a period of 3, a maximum value of 20, and a minimum value of 0. The function is a reflection of its parent function over the x-axis. Which function could be the function described? f(x)=10cos(3x)−10

3. ### Math

A quadratic function has a vertex of (3,-6) and the point (-1,10) lies on the graph of the function. Write the function in standard form. If you could help me with this question, that would be great. Thanks in advance.

4. ### Math I need help ASAP!

1) Identify the function rule shown in the table. Function Table n - 3, 4, 5, 6 y - 2, 1, 0, -1 a. y = 2 + n b. y = 5n c. y = 5 - n d. not enough information ** 2) What is the values of the function y = -2x - 4 for x = 0,1,2 and

1. ### calcius

Determine whether the relation represents a function. If it is a function, state the domain and range. {(-2, 1), (-1, -2), (0, -3), (1, -2), (3, 6)} a. function domain: {-2, -1, 0, 1, 3} range: {1, -2, -3, 6} b. function domain:

2. ### math, please somebody help me

Given the function f(x) = x4 , and suppose g(x) is some other function of x. (a) Then the function F(x) = (g(x))^4 is the same thing as ... a. f '(g(x)) · g'(x) b. f(g(x)) c. f '(x) d. f '(g(x)) e. g(f(x)) (b) By either the

3. ### Calculus (Continuity)

If the following function is continuous, what is the value of a + b? f(x) = {3x^2 - 2x +1, if x < 0 a cos(x) + b, if 0

4. ### Calculus

In order to express the function y=12^19x+7 as a composition of two functions the outer of which is an exponential function,we would let u be equal to the inner function, u=19x+7, and then write y as a function of u, as follows: