Increasing, Decreasing, Intercepts, Asymptotes, Etc.

[kply]

I have this problem:

Let f(x) = (x^2 + 3x - 1) / (x + 1). Find the following:

• all interepts.[/*:m:2bfphyar]
• the intervals where f(x) is increasing and decreasing[/*:m:2bfphyar]
• all relative extrema[/*:m:2bfphyar]
• the intervals where f(x) is concave up and down[/*:m:2bfphyar]
• all asymptotes[/*:m:2bfphyar]
• the graph of the function[/*:m:2bfphyar]

I think i have the intercepts which are:

. . . . .x-intercepts:

. . . . .x^2 + 3x - 1 = 0
. . . . .x = [-3 ± sqrt(9 + 4)] / 2
. . . . .x = [-3 ± sqrt(13)] / 2

. . . . .y-intercept: (0,-1)

For part b., I've done the following:

. . . . .f'(x) = [(x + 1)(2x + 3) - (x^2 + 3x - 1)(1)] / (x + 1)^2

. . . . .. . . .= (x^2 + 2x + 4) / (x + 1)^2

I am stuck there. I'm not able to get the value of x.

 

[tkhunny]

I am stuck there. I'm not able to get the value of x.

Have you considered that you may have th ecorrect solution? -- No Solution

Find where that numerator is zero. It's quadratic. Use the quadratic formula and PROVE that there are no Real solutions. This leads to the conslusion that there are no minima or maxima. Rewriting the original function a bit will help you see this:

\(\displaystyle f(x)\,=\,\frac{x^{2}+3x-1}{x+1}\,=\,x+2-\frac{3}{x+1}\)

You remember polynomial division from from Algebra, right?

This also helps you see the oblique asymptote quite clearly.