Issues with finding critical points on given intervals

[John45]

I need help with the follow finding there critical points
I know a critical point occurs when the derivative is equal to 0 or undefined.
If you could go step by step arrivals on to the answer I would greatly appreciate it

#1) sinx *cosx ; [0,2Pi]

I think the derivative is cos^2(x)-sin^2(x) but I cant seem to find when this is equal to 0

#2) x*ln(x/5) ; (0.1,5)

im not even sure if i did this right at all but for the derivative i got (x / (x/5) + ln(x/5) no clue again on what to do next

#3) 3t / ((t^2) + 1) ;[-2,2]

I think i got the derivative to (3t^2 + 3 - 6t) / (t^2 +1)^2

and i thought a critical point occured at t = -1

 

[Deleted member 4993]

I need help with the follow finding there critical points
I know a critical point occurs when the derivative is equal to 0 or undefined.
If you could go step by step arrivals on to the answer I would greatly appreciate it

#1) sinx *cosx ; [0,2Pi]

I think the derivative is cos^2(x)-sin^2(x) but I cant seem to find when this is equal to 0

Correct

\(\displaystyle cos^2(x)-sin^2(x) \ = \ 0\)

\(\displaystyle cos(2x) \ = \ 0\)

Continue...

#2) x*ln(x/5) ; (0.1,5)

im not even sure if i did this right at all but for the derivative i got (x / (x/5) + ln(x/5) no clue again on what to do next

\(\displaystyle \frac{5}{x}\cdot x \ + \ ln(\frac{x}{5}) \ = \ 0\)

\(\displaystyle 5 \ + \ ln(\frac{x}{5}) \ = \ 0\)

continue...

#3) 3t / ((t^2) + 1) ;[-2,2]

I think i got the derivative to (3t^2 + 3 - 6t) / (t^2 +1)^2

and i thought (thought ???? not calculated?) a critical point occured at t = -1 ....... Incorrect

Factorize the numerator and find its roots