Posted by **Ashley** on Monday, October 15, 2012 at 1:57pm.

f(theta)= 4sec(theta)+ 2tan(theta), 0 less than theta less than 2pi.

find all critical numbers

- Calculus -
**Steve**, Monday, October 15, 2012 at 3:11pm
where is f' zero or undefined?

f' = 4secθtanθ + 2sec^2 θ

= 2secθ(2tanθ+secθ)

so, critical numbers are where

secθ = 0 or is undefined,

2tanθ+secθ = 0 or is undefined

That help?

- Calculus -
**Ashley**, Monday, October 15, 2012 at 3:17pm
yes,i got 7pi/6 and 11pi/6 is that correct?

- Calculus -
**Reiny**, Monday, October 15, 2012 at 3:21pm
f(Ø) = 4secØ + 2tanØ

for Ø intercepts, f(Ø) = 0

4secØ + 2tanØ = 0

2/cosØ + sinØ/cosØ = 0

times cosØ

2 + sinØ = 0

sinØ = -2 , which is not possible

so the curve does not cross or touch the Ø axis

for y-intercept, let Ø = 0

f(0) = 4sec 0 + 2 tan 0 = 4 + 0 = 4

crosses the vertical axis at (0,4)

f ' (x) = 4 secØtanØ + 2sec^2 Ø

= 0 for max/min of f(Ø)

4/cosØ (sinØ/cosØ) = -2/cos^2 Ø

4 sinØ = -2

sinØ = -1/2

Ø = 7π/6 or Ø = 11π/6

if Ø = 7π/6 , f(7π/6) = 4(-√3/2) + 2/√3 = ....

if Ø = 11π/6, f(11π/6) = ...

I will let you finish the arithmetic

If you want the points of inflection, take the 2nd derivative, and set that equal to zero.

That will be quite a challenge.

- Calculus -
**Ashley**, Monday, October 15, 2012 at 3:28pm
Thank you!!

- Calculus -
**Steve**, Monday, October 15, 2012 at 3:34pm
Hmm. I was under the impression that critical numbers were numbers in the domain where f'=0 or f is not differentiable.

I think your math is great, Reiny, but maybe covers too much ground.

- Calculus -
**Anna **, Saturday, November 22, 2014 at 12:55am
f (0)=sinø + cos ø of(0)=3 f'(0)=4

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