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Question

Find the maximum and minimum values of y = tan x-2x.

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Solution

Given: fx =y=tan x-2xf'x =sec2 x-2For a local maxima or local minima, we must have f'x=0sec2 x-2=0sec2 x=2sec x=±2x=π4 and 3π4Thus, x=π4 and x=3π4 are the possible points of local maxima or a local minima.Now,f''x = 2 sec2 x tan xAt x=π4: f''π4 =2 sec2 π4 tan π4 = 4>0So, x=π4 is a point of local minimum.The local minimum value is given byfπ4 =tanπ4-2×π4 =1-π2At x=3π4: f''3π4 =2 sec2 3π4 tan 3π4 =-4<0So, x=3π4 is a point of local maximum.The local maximum value is given byf3π4 =tan 3π4-2×3π4 =-1-3π2

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