2 Points

$f(x) = \frac{lnx}{x}$

(a) $f'(x) = \frac{d}{dx}[\frac{lnx}{x}]$
Using the quotient rule:
$f'(x) = \frac{\frac{1}{x} \cdot x - lnx}{x^{2}}$
$f'(x) = \frac{1 - lnx}{x^{2}}$

For maximum, f'(x) = 0;
$\frac{1 - lnx}{x} = 0$
$lnx = 1, x = e$

(b) Deduce:
 $e^{x} \geq x^{e}, x > 0$

Soln: Since x = e is the greatest value, then f(e) ≥ f(x) > f(0)
$\frac{ln(e)}{e} \geq \frac{lnx}{x}$
$\frac{1}{e} \geq \frac{lnx}{x}$ , since ln(e) is simply equal to 1

Now, since x > 0, then we don't have to worry about flipping the signs when multiplying by x.
$\frac{x}{e} \geq lnx$
$x \geq elnx$
$x \geq ln(x^{e})$

Taking the exponential to both sides will cancel with the natural logarithmic function in the right hand side to produce:
$e^{x} \geq e^{ln(x^{e}})$
$e^{x} \geq x^{e}$ , as required.

3 0

3 years ago

The drama club is having a fundraiser selling novelty cups. Their goal is to make a profit of $1000. If they receive $3.50 profi

zlopas [31]

The answer is D because using division you would find how many cups you need to sell.

4 0

3 years ago

A. 2,464 cm^2 B. 2,364 cm ^2 C. 1,464 cm^2 D. 1,364 cm^2

Nastasia [14]

Answer:

c 1464 cm ^2 is the answer

5 0

3 years ago

A biologist is researching a newly-discovered species of bacteria. He puts 1,000 bacteria into what he has determined to be a fa

Masteriza [31]

The bacteria grow from 1,000 bacteria into 12,500 bacteria in 12 hours. The exponent function would be
:an= ao(1+r)^t12.500= 1,000(1+r)^121,000(12.5)= 1,000(1+r)^1212.5= (1+r)^121+r= 1.23427r= 0.23Number of bacteria after 20hr would be:an= ao(1+r)^tan= 1,000(1+0.23)^20an= 1,000(62.821 )an= 62,821

8 0

3 years ago

NEED ANSWERS ASAP, 100 POINTS! There are twenty classes at each of two middle schools. The number of students in each class at e

DiKsa [7]

Note: You missed to add the dot plots chart. As I found the chart after a little research. Thus, I am attaching it and based on that dot plot chart I am solving the question which anyways would clear you concept.

Answer:

'There are about 2 more students in each class at Oak Middle School than at Poplar Middle School' is the correct statement.

Step-by-step explanation:

From the diagram, it is clear that

The data set containing Poplar Middle School:

20 20 20 21 21 21 21 21 22 22 22 22 22 22 22 23 23 23 23 24

The mean of a data set is the sum of the terms divided by the total number of terms. Using math notation we have:

$\mu = 435/20$