Question

Must critical points be real numbers?

Answer 1

if you are working with real valued functions yes

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

oh sorry. missed that.

Answer 6

clear now yes?

Answer 7

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?

Answer 8

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

Answer 9

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

Answer 10

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.

Answer 11

Thank you.