Question

Find critical points and use interval notation to indicate where f(x) is increasing. For f(x)=(4+5x)/(3-2x)

Answer 1

http://www.twiddla.com/solved

Answer 2

What's the matter? Don't you want to know how to do it?

Answer 3

f'(x) = (5(3-2x) - -2(4+5x))/(3 - 2x)^2 = 23/(3 - 2x)^2

a squared number is always non-negative, hence f'(x) > 0 for all x except 3/2

hence f is increasing for all real x except 3/2

Answer 4

what about critical #'s?

Answer 5

there's also a slanted asymptote somewhere between 1 and 2

Answer 6

Okay, yeah, jamesm is right

Answer 7

critical point:
take the derivative
find out where it is 0
find out where it is undefined.
those are the critical points.

Answer 8

derivative is
\[\frac{23}{(3-2x)^2}\]

Answer 9

this is never zero because the numerator is a constant. it is undefined at
\[x=\frac{3}{2}\] but then again so is your original function so you get no information from this

Answer 10

increasing where the derivative is positive, and this is ALWAYS positive since numerator is a positive constant and the denominator is a square

Answer 11

there is no slant asympote so ignore that. the degree of the numerator is equal the degree of the denominator so you have a horizontal asympote at
\[y=-\frac{5}{2}\] the ratio of the leading coefficients

Answer 12

vertical asymptote at
\[x=\frac{3}{2}\]

Answer 13

so if you draw those two asympotes and remember that your function is always increasing you only have one picture you can draw.

Answer 14

satellite...it is always increasing where f is defined, yes? can't say it's always increasing in general