Increasing/Decreasing function, no max/min

[AquaGlass]

thanks!

 

[EnumaElish]

I obtained f ' (x) = [-(x^2) - 9]/[[(x^2) - 9]^2]
I found the critical numbers of 3 and -3.

Are these real roots, or did you mean 3i, -3i?

 

[EnumaElish]

+(x^2) - 9 has real roots 3 and -3.

-(x^2) - 9 = -((x^2) + 9) has the complex roots I stated.

 

[HallsofIvy]

I'm wondering why you find it hard to believe this has no max or min. This is a rational function that has vertical asymptotes at x= 3 and x= -3 (because the denominator factors as (x-3)(x+3)). If x< -3, all of x, x-3, and x-3 are negative so the fraction itself is negative. The graph goes from asymptotic to 0 for large negative x to going to -infinity as it approaches x= -3. If -3< x< 0, the fraction is positive. The graph goes from +infinity close to x= -3 through 0. If 0< x< 3, the fraction is negative and goes from 0 toward - infinity as x approaches 3. Finally, for x> 3 the fraction is negative and runs from near + infinity close to 3 to asymptotic to 0 as x goes to +infinity.
There are no turning points and no max or min.