`F(x) = 4 tan x-tan^(2)x+tan^(3)x,xnenpi+(pi)/(2)`

Options

A. is monotonically increasing
B. is monotonically decreasing
C. has a point of maxima
D. has a point of minima

Answer

1

Explanation

Here f(X) = 4 tan x -`tan^(2)x + tan ^(3)x`
`therefore f(X) =4 secT(2)x-2 tan x sect^(2)x+3 tan ^(2)x sec^(2)x`
`=sect^(2)x(4-2tanx +3tan^(2)x)`
`=3sec^(2)x{{:(tan^(2)x-2/3tanx+4/3):}}`
`=3sec^(2)x{{:(tanx-1/3)^(2)+((4)/(3)-(1)/(9))`
`=3sec^(2)x{{:(tan x-(1)/(3))^(2)+(11)/(9)}gtforall x`
Therefore f(X) is increasing for all x in domain