let f be the function defined for [pi/6,5pi/6] by f(x)=x+(sinx)^2a)find all values of x when f '(x)=1b)find the x coordinates of all minimum pointsc) find the x coordinates of all inflection points

⚑ Flag

f'(x)=1+sinxcosx=1 sinxcosx=0 sinx=o  in the interval given , the possible value is none. cosx=0 .In the given interval  , the possible value is x=/π/2 b) find the 2nd derivative  -2(sinx)^2+2 (cosx)^2=2cos2x when you equate it to 0, in the given interval , you get x=π/4. When taking values within the given interval on both the sides of π/4, like π/6 , and plug in 2 nd derivative you get positive value. Taking x=π/2 , you get negative. The point of inflection is π/4.The max is π/2 and the min is π/6