Question

Identify the intervals of increasing/decreasing, function's local extrema, by finding (a)1st deriv (b) sign table for f'(x)

Answer 1

\[\frac{ x-1 }{ x^2+5x+10 }\] That's the f(x). I already worked out that the first derivative (a) is

\[\frac{ -x^2+2x+15 }{ (x^2+5x+10)^2 }\]

I just need help trying to create a sign table which I think I need to find the domain for since I need critical points (I think?). But to find the domain I need to find what number makes the denominator equal zero (right?) But I don't think there's anything that would make the denominator equal zero so it doesn't have a domain (i'm guessing). So how would I find critical points to create a sign table so I can get local extrema?

Answer 2

The domain is the set of x values that we're allowed to use.
You are correct that this denominator has no zero, they're both complex roots.
So our domain would be all values for x, no zeroes that need to be excluded.

Setting the `numerator` equal to zero will allow us to find critical points.

Answer 3

\[\Large\rm -x^2+2x+15=0\]\[\Large\rm x=?\]

Answer 4

\[-(x^2-2x-15)\]
\[-(x+3)(x-5)\]
\[x= -3 , 5\]

And then how would I use those to find CP?

Answer 5

Or would -3 and 5 be the critical points I would use for the sign table

Answer 6

Those are your critical points.

Answer 7

Ahh. So just to make sure I understand. When I'm able to find the domain by making the denominator equal zero, I can use the x results there as critical points? And if I'm not able to get an answer from the denominator due to complex roots I turn to the numerator and solve it for zero?

Answer 8

I'm not sure what you're saying.
Lemme just explain a little further.
Yes, the denominator also gives us critical points.\[\Large\rm \frac{ -x^2+2x+15 }{ \color{orangered}{(x^2+5x+10)^2} }\]We would get critical points from this orange part.
But in this case we have no real solutions for that denominator, so no critical points.

Answer 9

Okay! Now I understand. Thanks!