Question

Find the critical numbers of f(t) = sin^2 t - cos t

Answer 1

What steps are involved in finding critical numbers?

Answer 2

1.) find the first derivative
2.) find numbers that make 1st derivative = 0 or undefined

did you find the 1st derivative?

Answer 3

ok.. so
\[f'(t) = 2\sin(t)\cos(t) + \sin(t)\]

Answer 4

Sorry,internet disconnected.
Let me show you where I've gotten to.

Answer 5

f prime of t = cos^2 t + sin t = 1 + sin t(1 - sin t)

Answer 6

You need chain rule for that first term
\[\sin^2(t) = [\sin(t)]^2\]

Answer 7

Oh my God! Thanks a bunch cruffo! I didn't properly differentiate sin^2(t), did I? :P

Answer 8

np : )

Answer 9

\[\left\{\{t\to 0\},\left\{t\to -\frac{2 \pi }{3}\right\},\left\{t\to \frac{2 \pi }{3}\right\}\right\} \]

Answer 10

\[\sin(t)(2\cos(t) + 1) = 0\]
sin(t) = 0 when t = any multiple of pi
2cos(t) + 1 = 0
2cos(t) = -1
cos(t) = -1/2
t = 2pi/3 + 2kpi or 4pi/3 +2kpi

Answer 11

\[t = \left\{k\pi, \; \pm\frac{2\pi}{3} + 2k\pi\right\}\]