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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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}.

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