Analyze the following rational function for...? Best answer for correct answer
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Analyze the following rational function for...? Best answer for correct answer

[From: ] [author: ] [Date: 11-08-15] [Hit: ]
---------------------I hope this helps!-Thanks for your kind words.This was a fairly involved question; so getting as many of the details written down took me awhile.Good luck!......
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(v) First Derivative Analysis.

Start with f(x) = (x^2 + 4x - 15) + 18/(x + 4) [for x ≠ -4, 1].
So, f '(x) = (2x + 4) - 18/(x + 4)^2.

Plainly f ' is undefined at x = -4, 1.
(For x = 1, we need not place it on a number line, because it is a removable discontinuity.)

Next, f '(x) = 0
==> (2x + 4) - 18/(x + 4)^2 = 0
==> x + 2 = 9/(x + 4)^2
==> (x + 2)(x + 4)^2 - 9 = 0
==> x^3 + 10x^2 + 32x + 23 = 0
==> (x + 1) (x^2 + 9x + 23) = 0
==> x = -1 (other two roots have nonzero imaginary parts).

This (and x = -4) breaks the the number line into three parts.
If x < -4, then f '(x) < 0 (try x = -1000).
If -4 < x < -1, then f '(x) < 0 (try x = -2)
If x > -1 (excluding x = 1, the hole), then f '(x) > 0 (try x = 0).

So, we have a local minimum at x = -1 (with value f(-1) = -12).

Otherwise, f is increasing for x in (-1, 1) U (1, ∞), and
f is decreasing for (-∞, -4) U (-4, -1).
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I hope this helps!

-
Thanks for your kind words.

This was a fairly involved question; so getting as many of the details written down took me awhile.

Good luck!

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