Derivatives and other related properties

[Guest]

OK we are doing a problem for class that our entire group is not certain of. Can you guys please help a.s.a.p. ? Thank you:

Let f be the function that is given by f(x) = (ax + b)/ (x^2 - c) and that has the following properties.

(I) THe graph of f is symmetric with respect to the y-axis.

(II) Lim f(x) as x ---> 2^+ = positive infinity

(III) F'(1) = -2

A) Determine the values of a b and c.
B) Write an equation for each vertical and each horizontal asymtote of the graph of f.
C) Sketch the graph of f in the xy-plane provided below.

We have tried solving for c and some think it is 4 but we are not sure. Thanks again for reading this and we hope to hear from you soon.

 

[skeeter]

Let f be the function that is given by f(x) = (ax + b)/ (x^2 - c) and that has the following properties.

(I) THe graph of f is symmetric with respect to the y-axis.

(II) Lim f(x) as x ---> 2^+ = positive infinity

(III) F'(1) = -2

A) Determine the values of a b and c.
B) Write an equation for each vertical and each horizontal asymtote of the graph of f.
C) Sketch the graph of f in the xy-plane provided below.

if f(x) is symmetric w/r to the y-axis, then f(x) is even ... f(-x) = f(x)

f(x) = (ax + b)/ (x^2 - c)
f(-x) = [a(-x) + b]/[(-x)^2 - c]
so ... what does "a" have to be?

lim{x->2 from the right} f(x) = +infinity ... c has to be ?

f'(x) = [(x^2 - c)(a) - (ax + b)(2x)]/(x^2 - c)^2
since f'(1) = -2 ...
-2 = [(1 - c)(a) - (a + b)(2)]/(1 - c)^2
if you get a and c from the above, then you can get b.

once you get (A), then (B) and (C) should be rather simple.