f(theta)= 4sec(theta)+ 2tan(theta), 0 less than theta less than 2pi.
find all critical numbers
yes,i got 7pi/6 and 11pi/6 is that correct?
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?
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.
Thank you!!
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.
f (0)=sinø + cos ø of(0)=3 f'(0)=4
To find the critical numbers of the function f(theta) = 4sec(theta) + 2tan(theta), we need to identify the values of theta where the derivative of the function is equal to zero or is undefined.
First, let's find the derivative of f(theta) with respect to theta. The derivative of sec(theta) is sec(theta) * tan(theta), and the derivative of tan(theta) is sec^2(theta). Using these derivative rules, we can find:
f'(theta) = 4 * sec(theta) * tan(theta) + 2 * sec^2(theta)
Next, we set the derivative equal to zero and solve for theta:
4 * sec(theta) * tan(theta) + 2 * sec^2(theta) = 0
Since the equation involves sec and tan, let's rewrite it in terms of sine and cosine:
4 * (1/cos(theta)) * (sin(theta)/cos(theta)) + 2 * (1/cos^2(theta)) = 0
Simplifying, we get:
4sin(theta)/cos^2(theta) + 2/cos^2(theta) = 0
Now, let's find a common denominator:
(4sin(theta) + 2) / cos^2(theta) = 0
To find the solutions, we need to set the numerator equal to zero:
4sin(theta) + 2 = 0
Subtracting 2 from both sides:
4sin(theta) = -2
Dividing by 4:
sin(theta) = -1/2
The solutions for theta that satisfy sin(theta) = -1/2 lie in the third and fourth quadrants of the unit circle. These values can be found by evaluating sin^(-1)(-1/2), which gives:
theta = 7π/6 and 11π/6
Therefore, the critical numbers of the function f(theta) = 4sec(theta) + 2tan(theta) are theta = 7π/6 and 11π/6.