Must critical points be real numbers?

if you are working with real valued functions yes

so for example if your function has derivative \[x^2+1\] it has no critical points

wouldn't 0 be a critical point in that case?

i meant that \[f'(x)=x^2+1\] so \[f(x)\] has no critical points

oh sorry. missed that.

clear now yes?

when simplified, the derivative I have is equal to \[13x ^{2}+8x+12=0\]. So it would likewise have no critical points because the roots are complex. Correct?

this has no zeros then it it always positive so your function is strictly increasing

so yes. if the zeros are complex, no critical points

Originally it had a factor of -4 so it was strictly decreasing, but yes. I see. I know that it has no extrema, but I was getting conflicting answers on whether it technically has critical points to define the intervals.

Thank you.

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