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Popular >

f(x)=4sec(x)+tan(x)

Solution

f (x) = 4 sec (x) + tan (x)

Solution

Period : 2 π

Domain : 2 πn \leq x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < \frac{3 π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < x < 2 π + 2 πn

Range : f (x) \leq 4 sec (π + arcsin (\frac{1}{4})) + \frac{1}{15} or f (x) \geq 4 sec (- arcsin (\frac{1}{4}) + 2 π) - \frac{1}{15}

Y Intercepts : (0, 4)

Asymptotes : Vertical x = \frac{3 π}{2} + 2 πn, x = \frac{π}{2} + 2 πn

Extreme Points : Maximum (π + arcsin (\frac{1}{4}) + 2 πn, 4 sec (π + arcsin (\frac{1}{4})) + \frac{1}{15}), Minimum (- arcsin (\frac{1}{4}) + 2 π + 2 πn, 4 sec (- arcsin (\frac{1}{4}) + 2 π) - \frac{1}{15})

+1

Interval Notation

Domain : [2 πn, \frac{π}{2} + 2 πn) \cup (\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn) \cup (\frac{3 π}{2} + 2 πn, 2 π + 2 πn)

Range : (- \infty, 4 sec (π + arcsin (\frac{1}{4})) + \frac{1}{15}] \cup [4 sec (- arcsin (\frac{1}{4}) + 2 π) - \frac{1}{15}, \infty)

Solution steps

Periodicity of

4 sec (x) + tan (x) : 2 π

Domain of

4 sec (x) + tan (x) : 2 πn \leq x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < \frac{3 π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < x < 2 π + 2 πn

Range of

4 sec (x) + tan (x) : f (x) \leq 4 sec (π + arcsin (\frac{1}{4})) + \frac{1}{15} or f (x) \geq 4 sec (- arcsin (\frac{1}{4}) + 2 π) - \frac{1}{15}

Axis interception points of

4 sec (x) + tan (x) :

Y Intercepts

: (0, 4)

Asymptotes of

4 sec (x) + tan (x) :

Vertical

: x = \frac{3 π}{2} + 2 πn, x = \frac{π}{2} + 2 πn

Extreme Points of

4 sec (x) + tan (x) :

Maximum

(π + arcsin (\frac{1}{4}) + 2 πn, 4 sec (π + arcsin (\frac{1}{4})) + \frac{1}{15}),

Minimum

(- arcsin (\frac{1}{4}) + 2 π + 2 πn, 4 sec (- arcsin (\frac{1}{4}) + 2 π) - \frac{1}{15})

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