Verify if the function f(x)=10secx+5tanx has critical numbers?

The critical values of the function are the roots of the
1st derivative, therefore, we'll have to compute the 1st derivative of
f(x).

f(x) = 10/cos x + 5sin x/cos
x

f(x) = (10+5sin x)/cos
x

We'll use the product rule to differentiate the function
with respect to x:

f'(x) = [5cosx*cosx + sinx(10+5sin
x)]/(cos x)^2

f'(x) = [5(cos x)^2+ 10sinx + 5(sin
x)^2]/(cos x)^2

We'll use the Pythagorean
identity:

(cos x)^2+ (sin x)^2 =
1

f'(x) = (5+ 10sinx)/(cos
x)^2

We'll cancel f'(x):

f'(x)
= 0

5+ 10sinx = 0

1 + 2sin x =
0

sin x = -1/2

x =
(-1)^k*arcsin (1/2) + k*`pi`

x = (-1)^k*(`pi` /6) +
k`pi`

The critical values of the function
belong to the set {(-1)^k*(` ` `pi` /6) + k`pi` / k`in`
Z}.